UNIVEF?SITY  OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL  ENGINEER.Ma 

BERKELEY.  CALIFORNIA 


CIYIL  ENGINEERING 

U.  of  C. 
ASSOCIATION  LIBRARY 


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UMivTTKai-rv  or  cAt-inroi'sNiA. 


OF 


Plane  Analytic  Geometry. 


A    TEXT-BOOK 


INCLUDING   NUMEROUS   EXAMPLES  AND   APPLICATIONS, 
AND   ESPECIALLY    DESIGNED   FOR    BEGINNERS. 


BY 

GEORGE    R.    BRIGGS, 

FORMERLY    TUTOR    IN    MATHEMATICS    IN    HARVARD    UNIVERSITY. 


SEVENTH  EDITION,  REVISED  AND  ENLARGED 
BY 

MAXIME    BOCHER, 

ASSISTANT  PROFESSOR  IN  HARVARD  UNIVERSITY. 

UNIVERSITY  OF  CAUFORNIA 

DEPARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY.  CALIFORNIA 

NEW  YORK: 

JOHN   WILEY   &    SONS. 

London  :   CHAPMAN  &   HALL,   Limited, 

1909 


Library 


Library 

Copyright,  1881,  1903, 

BY 

G.  R.  BRIGGS. 


Sdbprt  Brumuinnb  ani>  (Sompattg 


CIYiL  ENGINEERING 

U.  of  C. 

ASSOCIATION  LIBRARY 

PREFACE. 


The  design  of  this  book  is  to  present  the  elements  of 
Analytic  Geometry  in  a  simple  form,  for  the  use  of  beginners. 
With  this  object  in  view,  I  have  tried  to  make  the  demon- 
strations thorough,  sometimes  at  the  expense  of  brevity,  in  order 
that  the  student  may  never  feel  that  the  fundamental  truths 
have  been  in  part  assumed.  Therefore  it  may  often  seem  best 
to  the  teacher  to  present  special  cases  of  the  theorems  as  prepa- 
ration for  the  general  demonstrations  of  the  book. 

The  text  has  been  interspersed  with  a  large  number  of  exam- 
ples and  applications  to  familiar  theorems  of  Geometry,  often 
accompanied  with  the  answers  and  hints  for  the  solution,  to 
stimulate  the  interest  of  the  student  and  to  teach  him  early 
how  to  work  for  himself.  For  the  same  purpose  the  chapter  on 
Loci  has  been  introduced. 

Although  this  book  has  been  prepared  for  the  use  of  the 
Freshman  Class  in  Harvard  College,  it  is  hoped  that  it  may 
prove  useful    in    Schools  and  Academies,   and  to    any   student 

800299 


IV  PREFACE. 

having  a  fair  knowledge  of  Algebra  and  Trigonometry  who 
wishes  to  prepare  for  a  more  extended  course  in  Analytic 
Geometry. 

I  gladly  acknowledge  my  indebtedness  to  Prof.  Byerly  for 
his  kindness  in  reading  the  manuscript  and  for  valuable  sugges- 
tions. I  have  also  freely  used  the  works  of  Salmon,  Todhunter, 
Puckle,  and  others. 

G.  R.  BRIGGS. 

Cambridge,  April,  1881. 


The  changes  in  the  present  edition  consist  mainly  in  the 
addition  of  a  moderate  amount  of  new  matter.  Almost  all  of 
this  will  be  found  after  page  107,  and  in  particular  the  last  half 
of  Chapter  VIII  and  the  whole  of  Chapter  IX  are  new.  It  is 
hoped  that  in  its  present  form  the  book  will  be  found  to  cover 
the  ground  of  the  ordinary  first  course  in  analytic  geometry  as 
given  in  American  colleges  and  technical  schools,  and  that  it 
will  prove  serviceable  as  a  text-book  in  such  courses. 

Maxime  Bocher. 
Cambridge,  January,  1903. 


CONTENTS. 

♦ 

Chap.  Page 

I.   Introduction.  —  The  Point i 

II.   Locus  OF  AN  Equation,  Equation  of  a  Curve    .  12 

III.  The  Straight  Line 26 

IV.  Transformation  of  Coordinates 61 

V.   The  Circle 71 

VI.   Loci 96 

VII.  The  Conic  Sections 116 

VIII.  Diameters.     Poles  and  Polars 157 

IX.  The  General  Equation  of  the  Second  Degree  175 

Miscellaneous  Examples 184 


ELEMENTS  OF  ANALYTIC  GEOMETRY, 


CHAPTER   I. 
INTRODUCTION.  — THE   POINT. 

1.  The  reader  is  probably  familiar  with  algebraic  statements 
of  geometrical  theorems  which  concern  the  magnitude  of  lines  ; 
for  example,  where  a,  b,  and  c  represent  the  legs  and  hypotenuse 
of  a  right  triangle,  ^"^  +  i>^  =  c^  is  a  statement  in  algebraic  lan- 
guage of  the  Pythagorean  theorem. 

There  is  a  large  class  of  geometrical  theorems  which  concern 
the  position  of  lines  ;  for  example,  a  tangent  to  a  circle  is  per- 
pendicular to  the  radius  drawn  to  the  point  of  tangency.  Ana- 
lytic Geometry  enables  us  to  apply  algebra  to  this  class  of 
problems ;  and  as  we  advance  with  our  subject  we  shall  see 
that  this  use  of  algebra  shortens  our  work  and  generalizes  our 
reasoning. 

2.  We  shall  assume  that  the  reader  is  acquainted  with  the 
application  of  the  algebraic  signs  plus  and  minus  to  express  the 
upward  and  downward  directions  of  vertical  lines,  the  directions 
from  left  to  right  3.ndfrom  right  to  left  of  horizontal  lines,  and,  in 
general,  to  distinguish  between  the  opposite  directions  of  lines 
which  have  the  same  general  direction.  The  following  theorem, 
a  direct  result  of  the  distinction  between  positive  and  nega- 
Uve  lines,  will  be  useful  for  reference  :  — 


ANALYTIC   GEOMETRY. 


If  three  points,  Ay  B^  and  C,  lie  on  the  same  straight  line,  we 

may  write,  when  we  consider  the  directions  as  well  as  the  lengths 

of  lines, 

AB^  BC--AC, 

whatever  the  arrangement  of  these  points  on  that  line, 

3.  Analogous  to  the  theorem  of  §  2,  there  is  a  principle  for 
the  addition  of  angles  which  depends  upon  the  distinction  be- 
tween opposite  rotations  by  the  signs////j-  and  minus.  We  shall 
place  it  here  for  reference. 

If  three  lines,  O  A^  QBy  and  O  C,  pass  through  the  same  point 
O,  we  may  write 

AOB-{-  BOC=AOC, 

whatever  the  arrangemetit  of  the  three  lines  which  pass  through  O. 

4.  In  considering  problems  which  concern  the  position  of 
points  and  lines  in  a  plane,  we  must  have  some  fixed  objects  in 
the  plane  to  which  we  may  refer  the  position  of  each  point  and 
line.  There  are  several  combinations  of  fixed  objects  which  arc 
used  for  this  purpose,  the  most  common  being  two  straight  lines 
at  right  angles  to  each  other  and  indefinite  in  extent. 

In  the  figure,  X^  X  and 
F'  Fare  the  fixed  straight 
Hnes  intersecting  at  O.  The 
position  of  P  is  determined 
if  we  know  N P^  its  distance 
from  V  v.  and  MP,  its  dis- 
tance from  X'  Xy  together 
with  the  directiotts  of  these 
lines;  for  we  know  from 
geometry  that  P  is  the  only 
point  in  the  plane  at  a  dis- 
tance NP  to  the  right  of 
y  Fand  at  a  distance  MP 


p 

Y 

N 

P 

X'        M' 

II 

P 

0 

M               X 

III 
P 

N' 

THE   POINT.  3 

above  X^  X.  Let  P,  P\  F",  and  F^'^  be  the  corners  of  a  rec- 
tangle whose  middle  point  is  O,  and  whose  sides  are  parallel  to 
X'  X  and  K'  Y.  Then  recollecting  the  conventions  of  §  2,  if  we 
represent  the  length  IV F  by  a  and  the  length  M  F  hy  b,  the 
distances  of  F  from  Y'  F  and  X' ^  respectively  are  a  and  b; 
of  F\  -  a  and  b ;  ot  /^",  -  .^  and  -  b  ;  of  7^'",  ^  and  -  b. 

5.  We  see  that  two  elements  determine  the  position  of  each 
point.  These  are  the  straight  lines  which  indicate  by  their 
lengths  the  distances  of  the  point  from  X'  X  and  Y'  Y,  and  by 
their  signs  its  directions y^-^w  these  lines.  These  elements  are 
called  the  coordinates  of  the  point  which  they  determine  :  the 
one  which  indicates  the  distance  and  direction  from  K'  Y  is 
called  the  abscissa ;  the  other,  which  gives  the  distance  and 
direction  from  X^  X,  is  called  the  ordinate. 

In  the  figure,  N P,  or  its  equal  both  in  length  and  direction, 
O  M,  is  the  abscissa  of  F,  MP  is  the  ordinate  of  P ;  or  we  may 
say  that  the  abscissa  of  P  is  a,  and  the  ordinate  b ;  or,  more 
briefly,  that  the  coordinates  of  P  are  a  and  b,  the  abscissa 
always  being  named  first. 

The  fixed  lines  are  called  axes  of  coordinates,  and  we  speak 
of  X'  X  as  the  axis  of  x,  or  of  abscissas,  while  Y'  Y  is  called 
the  axis  of  j',  or  of  ordinates.  O  is  called  the  origin  of  coordi- 
nates, or  simply  the  origin. 

In  order  to  abbreviate  the  terms  abscissa  and  ordinate,  it  is 
customary  to  denote  them  by  the  letters  x  and  y  respectively  ; 
i.  e.  for  the  point  F,  x  —  a,  y  —  b ;  for  P\  x  z=z  —  a,  y  z=  b,  etc. 

6.  If  we  call  the  portions  of  the  plane  lying  in  the  angles 
XOY,  YOX',  X'OY',  Y'OX,  the  Jirst,  second,  third,  and 
fourth  quadrants  respectively,  we  readily  see  that  any  point  in 
the  first  quadrant  has  x  positive  and  y  positive  ;  any  point  in 
the  second  quadrant  has  x  negative  and  y  positive  ;  any  point 
in  the  third  quadrant  has  x  negative  and  y  negative  ;  any  point 


4  ANALYTIC    GEOMETRY. 

in  the  fourth  quadrah\f  has  x  positive  and  y  negative.  The  signs 
of  the  coordinates  of  a  point,  therefore,  show  in  which  quadrant 
the  point  lies,  while  the  magnitudes  of  the  coordinates  fix  the 
position  of  the  point  in  that  quadrant. 

7.  In  the  development  of  our  subject  we  shall  have  to  con- 
sider points  which  move  in  the  plane  and  others  which  are  sta- 
tionary. The  coordinates  of  a  moving  point  are  variables^  for 
each  assumes  in  turn  an  indefinite  number  of  values.  We  shall 
designate  a  moving  point  by  P^  its  coordinates  by  x  and  j,  and 
the  foot  of  its  ordinate  by  M. 

The  fixed  points  may  be  either  in  known  positions  or  un- 
known. We  shall  represent  known  points  by  P^,  P^,  P^,  etc., 
their  coordinates  by  (^1,  J\),  {^^.y-^,  (-^sj/s)?  ttc,  and  the  feet  of 
their  ordinates  by  M^,  M^,  J/o,  etc.,  respectively. 

The  points  whose  positions  are  unknown,  yet  determined  by 
the  conditions  of  the  problem,  we  shall  call  P\  P",  etc.,  their 
coordinates  (x\  y'),  (x",  y"),  etc.,  and  the  feet  of  their  ordinates 
M',  M",  etc.,  respectively. 

The  student  should  be  careful  to  accustom  himself  to  this 
notation,  as  it  is  very  important  not  to  confuse  the  moving 
points  with  those  which  have  determined  positions. 

Instead  of  designating  a  point  P^  as  having  coordinates  x^  =  a, 
ji  =  b,  we  often  write  the  point  (a,  b),  always  understanding  that 
the  abscissa  is  written  first;  e.  g.  the  points  (i,  2),  (3,  —  2),  are 
the  points  P^,  with  x^  —  1  and  y^  =  2,  and  P^,  with  x^  =  3  and 
}'2  =  -  2. 

8.  The  system  of  coordinates  described  in  §§  4-6  is  called 
rectangular,  to  distinguish  it  from  another  system  often  employed, 
in  which  the  axes  are  not  perpendicular  to  each  other,  but  may 
make  any  angle.  In  the  latter  system  the  abscissa  of  a  point  is 
measured  parallel  to  the  axis  of  x,  and  the  ordinate,  parallel  to 
the  axis  of  ^.     This  system  is  called  oblique;  and,  together  with 


THE    POINT. 


the  rectangular  system,  forms  the  general  class  of  rectilinear 
systems  in  distinction  from  the  po/ar  system,  which  will  be  men- 
tioned later. 

We  shall  always  suppose  coordinates  rectangular  unless  other- 
wise stated. 

g.  It  is  easy  to  represent  the  position  of  any  given  point,  in  a 
figure  drawn  to  any  scale.  This  is  called //^///;/^  or  cotistriidiiig 
the  point. 

Let  us  plot  the  points  (2,  3),  (-  2,  4),  (-  3,  -  2). 

The  first  point,  which 
we  will  call  P^^  is  at  a  dis- 
tance of  two  units  to  the 
right  of  O  V,  and  t/iree 
units  adove  O  X,  as  indi- 
cated by  its  coordinates. 
We  may  measure  off  the 
first  distance  from  O 
along  OX  to  M^,  which 
must  be  the  foot  of  the 
ordinate  of  P^.  Now  the 
ordinate  must  be  drawn 
upward    parallel   to   O  V, 

and  made  equal  to  three  ,        ,        ,^^^^^ 

^  I 1 1 1 1 1 

units  of  length.      Its  up-  012345 

per  extremity  is  P^,  the  required  position. 

To  plot  P^,  we  lay  off   O  M,  =  -  2  and  M.P^  =  4-     To   plot 

P^,  we  make   O  Af^  =  -  ^  and  M^P^  =  -  2.     In  a  similar  way 

any  point  may  be  plotted  or  constructed. 


M3 

p. 

Y 

f' 

■ 

I 

VI 2 

0                Ml                       X 

EXAMPLES. 

(i.)    Plot  the  points  (4,  -  3),  (6,  o),  (-  3,  o;,  (o,  4),  (o,  -  2), 
(0,0). 


(2.)    Draw  the  straight  line  whose  extremities  are  (—  2, 
(^,  -  3)- 


and 


ANALYTIC    GEOMETRY. 


(3.)    Draw  the  triangle,  the  vertices  of  which  are  the  points 
(-  4,-3).  (6,  i),  (4,  ii)- 

10.    We  will  now  seek  an  expression  for  the  distance  between 
two  given  points,  in  terms  of  the  coordinates  of  these  points. 

Let  Py.  with  coordinates  x^  = 
O  J/j,  }\  =  M^  P^,  and  P^,  with 
coordinates  x,-,  ~  O  M^,  y^  = 
M^Po,  be  the  given  points. 
Let  8  represent  the  distance 
between  P^  and  P^.  Through 
P^  draw  a  parallel  to  O  X,  and 
let  it  meet  M^P^  (or  this  line 
extended  if  necessary)  in  the 
point  R.     Now  we  know  that 


X' 


o"x["  M,  M2 

■^  N"^  j^J-^Ji/^P^PP,^  must  be  a  right  tri- 
angle, and  therefore  that  ^  2 

B''  =  P^'+  PP, (i) 

But  P^  P  is  equal  to  jW^  M,_,  the  opposite  side  of  a  rectangle, 
both  in  length  and  direction.  The  points  M^,  O,  and  M^  are  by 
construction  on  the  same  straight  line  X'  X,  and  therefore,  by 
§  2,  we  may  write 

P^A  =  Af,M,  =  Jf^O+  0M,=  OMo^-  OM^ 

=  X,  -  x^  ....  (2) 
Again,  P,  M^^,  and  P^  are  by  construction  on  one  straight  line, 
and  therefore,  by  §  2, 

PP^  =  PM,  +  M,P,  =  M,P,  -  M,P 
=  M,P,-  M^P^ 

=  y2-yv (3) 

because  M^P  and  M^P^  2ire  equal,  being  opposite  sides  of  a 
rectangle. 

Substituting  in  equation  (i)  the  values  of  P^  R  and  R P^  found 
in  equations  (2)  and  (3),  we  have 


THE   POINT.  7 

or  8  =  V(^2-^i)'  +  (j2->'iy' ; 

an  equation  which  answers  the  requirements  of  our  problem. 

We  may  interchange  J^^  and  J^^  in  the  value  of  8  just  found,  as 
by  so  doing,  though  we  change  the  signs  of  the  quantities  in  the 
parentheses,  we  do  not  change  their  squares. 

Note.  —  The  student  should  study  the  solution  of  this  problem 
with  great  care,  and  satisfy  himself  that  every  step  in  the  reason- 
ing is  applicable  to  all  the  modified  forms  of  the  figure  which 
result  from  taking  F^  and  P^  in  different  quadrants.  If  he  con- 
siders the  theorem  stated  in  §  2  until  its  full  meaning  becomes 
clear,  he  will  be  convinced  of  the  generality  of  the  solution  of 
this  problem.  It  is  important  that  he  be  so  convinced,  as  we 
have  here  illustrated  in  a  simple  form  a  method  of  reasoning 
which  will  be  often  employed  hereafter. 

EXAMPLES.* 
(i.)    What  is  the  length  of  the  straight  line  which  joins  (7,  -  2) 
with  (4,  -  3) .? 

Here  we  may  write 

•^1  =  7.    7i  =  -  2,     :^2  =  4,    ^2  =  -  3- 
Substituting  these  values  in  the  expression  for  8,  we  have 

8  =  V[4-7p  +  [-3-(-2)r  =  '^T^WTJpry 

=  V'  10         Ans. 
(2.)    Find  the  distance  between  (2,  3)  and  (-  2,  8). 

Ans.   'v/zfT- 
(3.)    Find  the  length  of  the  line  given  in  §  9,  Ex.  (2). 

Atis.  2  V^. 
(4.)    Show  that  the  lengths  of  the  sides  of  the  triangle  men- 
tioned in  §  9,  Ex.  (3),  are  in  the  same  ratio  as  V26,  V29,  V65' 
(5.)    Show  that  the  distance   of  a  given  point,  P^,  from  the 
origin  is  \/ x^  +  y^. 

*  In  solving  examples,  the  student  should  construct  figures,  plotting  each  point. 


8 


ANALYTIC   GEOMETRY. 


(6.)  Calculate  the  lengths  of  the  sides  of  the  rectangle  whose 
vertices  are  the  points  (a,  b),  (-  a,  b),  (-  a,  -  d),  {a,  -  b)  ; 
show  that  the  vertices  are  equidistant  from  the  origin ;  prove 
that  the  diagonals  are  equal. 

II.  Let  us  find  the  coordinates  of  a  point  F'  so  situated  on 
^-\e  straight  line  which  passes  through  two  given  points,  F^  and 
F.,,   that  its  distances  from  these  points  respectively  are  in  a 

7  ^1 

giVen  ratio,  k  =  — -. 


B}^  the  conditions  of 

-  ,  ,       F.F^     m, 

the  problem 

are 


F^F'  ?n„ 
where  77t^  and 
ajty  two  quafitities  and 
have  no  connection 
with  the  points  M^ 
and  M^.  Through  F^ 
draw  a  line  parallel  to 
O  A^and  cutting  M^  F^ 
and  M^F^  (or  their 
extensions)  in  R  and  S  respectively.  Since  AI^  F^  and  M^  F,^  are 
necessarily  parallel,  the  triangles /^^ i? 7^^  and  F.^SF'  are  always 
similar,  and  therefore 


FF' 


F,F' 


S  F'~  F,Ff~  n, 
Now,  by  §  2, 

FF'  =  M^M'  =  M,0+  OAF 


OMf  -  OM. 


(I) 


and  similarly 


SF^  =  M.JF  ^x'  -x^. 


Substituting  these  values  of  F  F'  and  S F'  in  equation  (i), 
A^e  have 


X'  -  x„ 


THE    POINT.  9 

Solving  for  x', 

m^  —  m^     ' 
which  gives  the  value  of  the  unknown  abscissa  of  P^  in  terms  of 
known  quantities. 

Similarly  we  may  find  y  ;  for 

P^^S  ~  P^P'-  m^ ^  ^ 

Now,  by  §  2, 

P^P  =  P^M^  +  M^P=  M,R  ~  M^P^ 
=  MiP'-M^P^ 

Again, 

P^S  =  M^S-M^P^ 

=  y-yr 

Substituting  in  (2),  we  have 


y  -  yx 
y  -  y-i 

m^ 
m^ 

m,y,  - 

^hy^ 

or  v' 

which  expresses  y  in  terms  of  known  quantities. 

We  have  now  found  both  x^  and  y\  and  therefore  solved  our 
problem. 

N.  B.  By  the  conditions  of  the  problem  the  distances  of  P^ 
fro77i  P^  and  P.^  have  the  ratio  k.  As  both  lines  are  to  be  meas- 
ured toivard  P\  they  will  have  the  sa^ne  direction  when  P'  is  not 
between  P^  and  P^ ;  and  opposite  directions  zvhen  P'  lies  betiveen 
P^  and  P^.  In  the  first  case  the  ratio  k  must  ho.  positive ;  in  the 
second,  negative. 

The  student  should  carefully  notice  that  in  the  first  case  the 

R  P^  P  /? 

ratios       ^  and  ^      of  equations  (i)   and  (2)  are  necessarily 

positive ;  in  the  second,  negative. 


lO  ANALYTIC    GEOMETRY. 

Corollary.     A  common  case  of  this  problem  is  that  in  which 
P^  bisects  the  line  /\  P^.     In  this  case 
P^P^---P^P\ 


and  the  ratio 


we  mav  therefore  write 


^  =  ^  =  -,j 

m^ 


and,  with  this  simplification,  the  coordinates  of  P^  become 


the  formulas  for  bisection. 


EXAMPLES. 

(i.)  On  the  straight  line  through  (2,  3)  and  (8,  -  12),  find 
a  point  such  that  its  distances  from  the  given  points  respectively 
shall  be  in  the  ratio  —  \. 

Here 

.Ti=.2,     J^i  =  3,      :x:2  =  8,     JJ'2  =  -I2. 
m^—  —  I*,     ;;?2  =  2. 
Substituting  these  values  in  the  expressions  for  x^  and  y\  we 
have 

,_2x2-(-i)8       4  +  8 


(-0         3 


=  4 


v'  -  ^   >^  3  -  (-  i)  (-    12)  _  6-12  _  _ 
■^    ~  2  _  (-  .)  ~        3        ~ 

The  required  point  is,  therefore,  (4,  —  2). 

Note.  This  problem  may  also  be  stated  as  follows.  Find  the 
point  of  trisection,  nearest  (2,  3),  of  the  line  which  joins  (2,  3) 
with  (8,  -  12). 

*  When  k  is  negative,  it  is  immaterial  which  of  the  quantities  nix  and  7n„  is  made 
negative.  This  will  be  readily  seen  by  noticing  that  when  the  signs  of  both  w,  and 
Wo  are  changed  in  the  formulas  for  x'  and  j',  the  signs  of  the  numerators  and  denomi- 
nators are  reversed,  and  therefore  the  signs  of  the  fractions  remained  unchanged. 


THE    POINT.  II 

(2.)  Construct  the  points  and  line  given  in  the  previous 
example,  and  verify  the  results  by  calculating  the  distances 
between  the  given  points  and  the  point  (4,  —  2). 

(3.)  Find  the  middle  point  of  the  line  which  joins  (2,  3)  and 
(10,  7).  Ans.\6,  5). 

(4.)  Find  that  point  of  the  Hne  joining  (-  4,  5)  with  (i,  2), 
whose  distances  from  these  points  respectively  are  in  the  ratio 
3  :  2.  Ans.  (11,-  4)- 

(5.)  A  line  is  dr3.wu/ropi  (-  2,  —  5)  fo  (3,  2)  ;  to  what  point 
must  it  be  extended,  in  the  same  direction,  that  its  length  may 
be  doubled  ?  to  what  point  that  it  may  be  trebled  ? 

Ans.  (8,  9);  (13,  16). 

(6  )  Find  the  middle  points  of  the  sides  of  the  triangle  men- 
tioned in  §  9,  Ex.  (3).  Ans.  (i,  -  i),  (5,  6),  (o,  4). 

(7.)  In  the  same  triangle,  find  the  coordinates  of  the  points 
of  trisection  (remote  from  the  vertices)  of  the  lines  which  join 
the  vertices  to  the  middle  points  of  the  opposite  sides. 

Ans.  (2,  3). 

(8.)  Solve  the  same  problem  when  the  vertices  of  the  triangle 
are  (x^,  j\),  {x^,  7,) ,  (^3  y^)  ;  show  that  your  results  prove  a  well- 
known  theorem  of  Geometry. 

(9.)  In  any  right  triangle,  the  straight  line  drawn  from  the 
vertex  of  the  right  angle  to  the  middle  of  the  hypotenuse  is 
equal  to  one  half  the  hypotenuse.  Prove  this  theorem  by 
means  of  §§  10,  11. 

Suggestion.  —  Let  the  vertex  of  the  right  angle  be  the  origin 
and  the  legs  of  the  triangle  the  axes  of  coordinates.  The  ver- 
tices of  the  triangle  may  then  be  expressed  as  the  points  (o,  o), 
{x„  o),   {o,y^). 

(10.)  Show  that  the  straight  lines  joining  the  middle  points 
of  the  adjacent  sides  of  any  quadrilateral  form  a  second  quad- 
rilateral whose  perimeter  is  equal  to  the  sum  of  the  diagonals  of 
the  first. 


CHAPTER  11. 


LOCUS   OF    AN    EQUATION. 


12.  We  have  seen  that  the  position  of  a  point  in  a  plane  is 
completely  determined  when  we  know  its  abscissa  and  ordinate  ; 
i.  e.  when  we  know  its  distances  fron^  Q  j\ind  OX.  Of  course, 
if  no  information  is  given  concerning  the  coordinates,  x  and  y, 
of  a  point  7^,  the  point  may  have  any  position  in  the  plane. 
Now  we  shall  often  have  to  consider  points  concerning  whose 
coordinates  we  know  so?7zet/iing,  yet  not  enough  to  determine 
them.  In  this  case  we  shall  find  that  the  point  can  no  longer 
occupy  any  position  in  the  plane,  nor  is  it  limited  to  a  single 
position,  but  that  //  is  coJifined  to  a  series  of  positions,  any  one 
of  ivhich  it  may  take  and,  at  the  sa7ne  time.,  have  its  coordinates 
satisfy  the  conditions  imposed  upon  them. 

To  illustrate :  let  us 
suppose  that  we  know 
the  abscissa  of  a  point  to 
be  zero  and  have  no  in- 
formation concerning  its 
ordinate  ;  i.  e.  we  have 
given  only  x  ^  o. 

This  equation  states 
that  the  distance  of  F 
from    the    axis    of   y   is 


Y 

a 

B 

a 

o 

X 

c 

D 
A 

nothing 


or,    in    other 


words,    that    /^   lies    on 


LOCUS    OF    AN    EQUATION. 


13 


O  V;  but  as  we  do  not  know  the  value  of  y,  the  ordinate  of  F, 
which  expresses  its  distance  from  OX,  the  point  may  lie  afiy- 
where  on  O  Y.  Thus  we  see  that  F  may  move  anywhere  along 
O  V,  and  yet  always  have  the  equation  x  =  o  true  for  it. 

Likewise  the  single  equation  y  =  o  states,  in  algebraic  lan- 
guage, that  the  point  F,  of  which  y  is  the  ordinate,  must  lie  on 
O  X,  but  allows  it  to  take  any  position  on  OX. 

Again  :  if  we  have  given  x  =  a,  we  know  that  the  distance  of 
the  point  F  from  O  Fmust  be  a  units  of  length ;  F  must,  there- 
fore, lie  somewhere  on  the  line  AB  parallel  to  O  Fand  at  the 
distance  a  from  it ;  but  it  may  occupy  any  position  on  A  B. 

In  the  same  way  the  single  equation  y  —  —  a  indicates  that  P 
may  lie  anywhere  on  CD,  parallel  to  (9^  and  a  units  below  it. 

Let  us  next  examine  the  equation  x  —  y.  We  see  at  once  that 
this  equation  does  not  determine  x  and  j,  the  coordinates  of  F, 
but  that  it  does  show  us  that  the  coordinates  of  F  are  always 
equal  to  each  other  both  in  numerical  value  and  algebraic  sign  ; 
or,  in  other  words,  that  the  point  must  be  just  as  far  from  one 
axis  as  from  the  other,  and  that  its  directions  from  the  axes  are 
either  both  positive  or  both  negative. 

The  point  will  satisfy 
both  of  these  conditions 
if  it  is  anywhere  on  the 
line  A  B  which  bisects 
the  angle  X  O  Y,  and  it 
cannot  fulfil  them  unless 
it  is  a  point  of  A  B.  If 
the  point  lies  on  the 
part  of  the  line  in  the 
angle  X  O  Y,  its  coor- 
dinates are  equal  and 
positive  ;  if  on  the  part 
in  the  angle  X'  O  Y',  they  are  equal  and  negative. 


14  ANALYTIC    GEOMETRY. 

If  the  coordinates  of  P  are  connected  by  the  equation  x  =  —  y^ 
we  readily  see  that  P  must  lie  somewhere  on  the  line  CD  which 
bisects  the  angle  Y  O  X\ 

13.  The  simple  examples  of  §  12  illustrate  the  fact  that  any 
single  equation  connecting  x  and  7,  the  coordinates  of  P^  though 
not  determining  these  quantities,  states  some  law  which  governs 
their  changes,  and,  so,  confines  P  to  some  series  of  positions  in 
the  plane.  This  series  of  positions  which  a  point  can  occupy 
while  its  coordinates  satisfy  a  given  equation  is  called  the  locus 
of  that  eqiiatiofi. 

We  may  also  define  the  locus  of  an  equation  as  follows  :  — 
The  locus  of  an  equation  is  the  path  of  a  point  which  moves  in 
the  p  la  fie  so  that  at  each  ifistant  its  coordinates  satisfy  that  equatio?t 
when  substituted  therein  for  x  and y^  the  variables  in  that  equation. 

14.  Looking  at  this  relation  between  an  equation  and  a  line 
from  a  somewhat  different  point  of  view,  we  notice  that  the  line 
A  B  m  the  last  figure  has  the  following  geometric  property :  — 

Each  point  in  A  B  \s  equally  distant  fro?n  O  Y  and  O  X;  a 
property  which  (remembering  the  distinction  between  positive 
and  negative  directions)  no  other  point  in  the  plane  possesses. 
Now  the  distances  of  a  point  from  O  Kand  O  Xare  the  coordi- 
nates x  and  y  of  that  point,  and  therefore  the  equation  x  =y 
expresses  algebraically  the  distinctive  property  of  points  on  A  B, 
or,  as  we  may  say,  of  the  line  A  B.  This  equation,  therefore, 
is  called  the  equation  of  AB. 

We  may  define  the  equation  of  a  curve  *  as  follows  :  — 
The  equation  of  a  curve  is  the  expression^  in  an  equation.,  of  the 
relation  which  exists  between  the  coordinates  of  every  point  of  that 
curve,  and  of  no  other  points. 

15.  This  relation  between  a  curve  and  its  equation,  a  relation 

*  The  word  curve  is  used  in  Analytic  Geometry  to  indicate  any  geometric  locus, 
whether  a  curve,  a  straight  line,  or  an  isolated  point. 


LOCUS    OF    AN    EQUATION.  I5 

SO  intimate  that  either  may  be  taken  as  a  representative  of  the 
other,  is  the  foundation  of  our  subject,  and  cannot  be  too  care- 
fully studied. 

The  development  of  the  subject  will  be  natural  and  easy  to 
the  student  who  has  so  thoroughly  mastered  this  idea  that  he 
never  for  an  instant  loses  sight  of  it ;  while  to  one  who  over- 
looks it,  or  whose  conception  of  it  is  vague,  the  whole  subject 
must  be  unintelligible  and  difficult. 

16.  In  §  12  we  have  determined  the  loci  of  several  equations 
by  observing  the  geometric  properties  which  are  stated  in  alge- 
braic  language  in  those  equations.  This  method  is  useful  for 
simple  equations,  where  the  geometric  property  is  a  simple  one. 
For  instance,  we  may  obtain  the  locus  of  x'^  +  y^  =  a^  in  this 
way  ;  for,  by  §  10,  Ex.  5,  we  know  that  x'-^  +  y^  represents  the 
square  of  the  distance  of  the  point  P,  which  has  x  and  y  for  its 
coordinates,  from  the  origin.  Our  equation  therefore  states  that 
/'moves  in  the  plane  so  that  the  square  of  its  distance  from  O  is 
constant  and  equal  to  a'.  The  distance  of  J^  from  O  is  therefore 
constant  and  equal  to  a,  which  can  only  be  the  case  when  /^ 
moves  on  the  circumference  of  a  circle  with  a  for  a  radius  and  O 
for  a  centre.  Consequently  this  circumference  is  the  locus  of 
the  equation  x'  +  y'^  —  d^. 

i^j.  There  is,  however,  another  method  of  determining  ap- 
proximately the  locus  of  an  equation,  founded  on  the  fact  that 
the  locus  is  the  assemblage  of  all  points  whose  coordinates  sat- 
isfy that  equation.  We  can  often  find  enough  of  these  points  to 
show  the  general  form  of  the  locus  ;  for  we  must  bear  in  mind 
that  a  curve  drawn  at  random  (as  in  the  figure)  is  not  the  locus 
of  any  equation.  The  equation  expresses  some  /aw  which 
g07>erjis  the  changes  of  the  coordinates  of  P^  and  consequently 
controls  the  patn  of  P  in  the  plane  ;  therefore  the  locus  of  the 
equation  must   be  a  curve  governed  by  some  law,  and  cannot 


i6 


ANALYTIC    GEOMETRY. 


be  a  line  drawn  at  ran- 
dom. Hence  the  general 
form  of  the  curve  will  be 
represented  by  the  posi- 
tions of  points  upon  it, 
chosen  at  reasonable  in- 
tervals. 

This  method  of  deter- 
mining a  curve  by  points 
is  called //(?///;;^  the  curve. 
Let  us  plot  the  curve  whose  equation  \s  x  —  y  -\-  2  =  o. 
If  we  choose  any  value  for  x,  we  can  find  a  value  for  7  which, 
together  with  the  chosen  value  of  x,  will  satisfy  the  equation. 
These  values  are  therefore  the  coordinates  of  one  point  of  the 
locus  of  the  equation.  If  we  choose  :r  =  i,y  must  equal  3  in 
order  that  x  —  y  +  2  may  equal  o :  therefore  (i,  3)  is  a  point  of 
the  required  locus. 

Similarly   (2,  4),   (4,  6),  (8,  10),   (o,  2),   (-  i,  i),   (-  2,  o), 

(-  3.  -  0.  (-  5, 
-3)y  (-8.  -6),  are 
points  of  the  lociis. 
Callingthese  points 
jP^,  P,,  etc.,  to  /",o, 
and  constructing 
them  as  indicated 
by  the  figure,  we 
observe  that  they 
all  lie  on  a  straight 
line  ;  and  from  this 
we  infer  that  the 
locus  of  the  equa- 
tion X  —  y  -\-  2  —  o 
is  the  straight  line 


/Pio 


LOCUS    OF    AX    EQUATION.  1 7 

which  cuts  O  i'two  units   above  (9,  and  OX  two  units  to  the 
left  of  O. 

We  shall  arrive  at  the  same  result  if  we  use  the  method  of 
^  i6.  For  the  equation  may  be  written  _v  =  .v  —  2.  which  inter- 
preted in  geometric  language  means  that  ever}-  point  of  the  locus 
has  an  ordinate  two  units  grctitcr  than  its  at'sa'ssa  :  or,  in  other 
words,  that  t/ie  distance  of  each  point  from  O  X  is  greater,  by  two 
I/nits,  than  its  distance  from  O  V.  Xow  in  §  12  we  showed  that 
the  straight  line  bisecting  X O  i'had  all  its  points  equally  dis- 
tant from  the  two  axes  ;  the  required  locus  is  therefore  parallel 
to  that  bisector,  and  at  a  distance  of  two  units  from  it  when 
measured  parallel  to  O  )'.  It  will  be  seen  that  this  result  agrees 
with  the  line  which  we  have  just  plotted. 

Let  us  plot  the  locus  of  y  —  4  x. 

In  this  equation  : 
if  .V  =  o,  y  =  o] 

if  .V  r=  I,  r  =  i  2  : 

if  .r  =  2,y  =  ±  2  \  2  =r  ±  2.S2  ; 

if  x  =  2,.y  =  -^2  \  3  =^13.46; 

if  .V  =  4.  V  =  =  4  : 

if  a-  =  5.j=:=  2  \  5  =^4.48; 

if  a-  =  6,  V  =  i  2  \  6  =  db  4.90 ; 

it  x=.'j,y  =  ±2\  7  =  ^5.30; 

if  A-  =  S,j—i4  \  2  =±5.64; 

if  X  =r  9,  _v  =  =  6  ; 

and  so  forth. 

We  must  also  notice  that  any  negative  value  of  x  \-ields  imas:- 
inar}-  values  for  v.-  hence  we  conclude  that  there  are  no  points 
of  this  locus  with  negative  abscissas,  i.  e..  no  points  to  the  left  of 
O  y.  Again,  each  abscissa  has  two  ordinates.  equal  in  length 
but  opposite  in  direction,  either  of  which  may  accompany  it  and 
satisfy  the  equation  :   from   this  we  infer  that  for  ever}-  point 


iS 


ANALYTIC    GEOMETRY. 


above  (7 X  there  is  a  point  symmetrically  situated  below  OX, 
or  we  may  say  that  the  curve  is  symmetrical  with  respect  to  O  X. 

It  is  also  evident  from  the  equation,  that  the  larger  x  is,  the 
larger  _y  must  be  ;  there  can  be  no  limit,  therefore,  to  the  extent 
of  the  curve  to  the  right  of  O  Y;  and  it  will  go  on  forever  in 

this  direction  both 
above  and  below 
(9X,  gradually  re- 
ceding from  both 
axes. 

Plotting  the 
points  which  we 
-  have  found,  we 
see  that  the  form 
of  the  curve  is 
well  represented 
except  between 
the  points  (o,  o) 
and  (i,  ±  2).  it 
will  be  well,  there- 
fore, to  find  more  points  in  this  interval. 

If  ^  =  1,  J  =  ±  I  ; 

if  X  —  \,  y  —  ±  \/2  =  ±  1. 41. 

Plotting  these  points  in  addition  to  those  already  found,  the 
curve  may  be  represented  with  tolerable  accuracy  by  a  line 
drawn  through  them,  and  having  the  form  indicated  by  these 
points.  As  we  have  shown,  it  will  extend  indefinitely  to  the 
right. 

EXAMPLES. 

Plot  the  curves  whose  equations  are  :  — 

y\i.)    2.T  +  3j-6  =  o;         (2.)    5^+j'  +  2=:o; 
(3.)    jc2+/  =  25;  ^(4.)    4Jt2  +  9/==  144; 


LOCUS    OF    AN    EQUATION.  I9 


(5-)  4^^-9^  =  36; 

{6-)  y^x'; 

(7.)/-^^; 

(8)  x^-f  =  o; 

(9.)  y  =  sin  x; 

(10.)  y  =  tan  x; 

(11.)  y  =  \og^^x; 

{i2.)y=2\ 

(13.)   Show  that  equation 

4-^    +  SX  +  2  =  o 
has  no  locus. 

(14.)   Show  that  the  locus  of  the  equation 

x'  +  2y'  =  o 
consists  of  a  single  point. 

18.  The  distance  fropi  the  origip  along  the  axis  of  .v  to  a 
point  where  any  curve  cuts  the  axis  is  called  an  intercept  of  that 
curve  on  OX.  As  a  curve  may  cut  the  axis  of  x  in  more  than 
one  point,  it  may  have  more  than  one  intercept  on  OX;  each, 
however,  is  measured  from  O.  In  the  same  way,  a  curve  has 
one  or  more  intercepts  on  6^  Y. 

The  intercepts  of  a  curve  may  be  easily  found  from  its  equa- 
tion ;  for  it  is  evident,  from  the  definitions  just  given,  that  the 
intercept  on  O  X\?>  the  abscissa  of  the  point  where  the  curve  cuts 
O  X,  and  the  intercept  on  O  V,  the  ordinate  of  the  point  where 
the  curve  cuts  O  Y.  Let  A  represent  the  first  of  these  points, 
and  B  the  second.  The  coordinates  of  A  may  be  represented  by 
X;^  and  _)'a  ;  those  of  B  by  x^  and  y^.  Now  since  ^  is  a  point  of 
the  curve,  its  coordinates  must  satisfy  the  equation  of  the  curve, 
when  substituted  therein  for  x  and  y ;  moreover,  y,,  =  o,  because 
^  is  a  point  of  OX.  Making  the  substitutions,  Xj^  is  the  only 
unknown  quantity  in  the  equation,  and  can  be  found  by  solving 
the  equation.  If  there  are  two  or  more  values  for  x^^,  the  curve 
has  two  or  more  intercepts  on  OX,  —  one  for  each  value  of  x^^. 

Similarly  the  intercepts  on  O  Fmay  be  found  by  substituting 
x^  and  Jb  for  x  and  v  in  the  equation,  and  then  making  x^  =  o. 

The  resulting  values  of  y^  are  the  required  intercepts. 


20  ANALYTIC    GEOMETRY. 

EXAMPLES, 
(i.)  Calculate  the  intercepts  of  the  curve  whose  equation  is 
2  X  —  ^y  —  6=0 (i) 

Writing  x^  for  x  and  Va  forjy,  in  equation  (i),  we  have 

2  Xa  —  3}'a  —  6  =  0 (2) 

Substituting  in  (2)  the  value  Ja  =  o,  we  have 

•^A  =  3- 
Again  ;  to  find  the  intercept  on  O  Y,  we  have 

2  .^B  —  3  ^B  —  6  =  o, 
and  Xb  =  o  'y 

therefore  j;b  =  —  2« 

The  curve  therefore  cuts  OX  three  units  to  the  right  of  the 
origin  and  O  Y  two  units  below  the  origin. 
(2.)    Find  the  intercepts  of  the  curve 

X  +  3/rr  27. 

The  intercept  on  O  X  may  be  found  as  in  the  last  example, 
and  is  27.  To  find  the  intercepts  on  O  Y,  we  substitute  x^  and 
y^  in  the  equation  of  the  curve,  and  obtain,  after  making  x^  equal 
to  zero,  ,J^'^  'il' 

3Jb^  =  27,  ^ 

and 

JJ^B    =   ±    3. 

This  curve,  therefore  cuts  O  Y  twice ;  once,  three  units  above 
O,  and  again,  three  units  below  O, 

Note.  —  In  practice  we  need  not  substitute  Xj,  and  y^,,  or 
ATg  and  j^B  in  the  equation.  We  shall  obtain  the  same  result  if 
we  first  put  jf  =  o  in  the  equation,  and  find  the  corresponding 
values  of  x ;  then  make  :v  =  o  in  the  original  equation,  and  find 
the  corresponding  values  of  jj'. 

The  results  are  the  intercepts  on  6>Xand  O  K  respectively. 

We  will  now  seek  the  intercepts  of  the  locus  of 
y"  ^\x. 


LOCUS   OF   AN    EQUATION,  21 

Makings  =  o,  we  have 


and 


o  =  /^x, 


X  =  o. 
This  shows  that  the  intercept  on  OX  has  no  length,  and  that 
the  curve  passes  through  the  origin. 

In  like  manner,  when  a;  =  o  in  the  equation,  we  have 

y  —  o. 
This,  too,  shows  that  the  curve  passes  through  the  origin ;  as 
it  must  cut  O  V  a.t  d.  point  whose  distance  from  O  is  zero. 

ig.  We  may  observe  that  a  curve  will  always  have  its  inter- 
cepts equal  to  zero,  and  therefore  will  pass  through  the  origin,  when 
every  term  in  its  equation  {reduced  to  the  simplest  form)  cotitains 
either  x  or  y  or  both.  For,  if  this  is  the  case,  when  we  substitute 
for  X  and  y  in  the  equation  the  coordinates  of  the  origin  (o,  o), 
each  term  vanishes  and  the  equation  is  satisfied.  This  shows 
that  the  origin  is  a  point  of  the  curve. 

On  the  other  hand,  if,  when  the  equation  is  in  its  simplest 
form,  it  contains  a  term  in  which  neither  x  nor  y  occurs,  the 
curve  cannot  pass  through  the  origin.  For,  in  this  case,  when 
we  substitute  o  for  x  and  o  for  j,  the  equation  becomes 

C  =  o, 
where  C  represents  a  constant  which  is  not  o.     Therefore  the 
equation  is  not  satisfied  by  the  coordinates  of  the  origin,  and  the 
curve  cannot  contain  that  point. 

EXAMPLE. 
Calculate  the  intercepts  of  each  of  the  curves  whose  equations 
are  given  in  the  examples  of  §  17. 

h^  10*  If  we  wish  to  find  the  intersection  of  two  curves,  given  by 
their  equations,  we  have  only  to  consider  that  the  point  of  inter- 
section of  two  curves  is  a  point  of  ecuh  curve,  and,  therefore, 


22  ANALYTIC    GEOMETRY. 

that  its  coordinates  must  satisfy  each  equation,  when  substituted 
for  X  and  j^.  If  P^  represents  the  required  point,  we  shall  have, 
after  substituting  its  coordinates  x^  and  y^  in  the  equations  of 
the  two  curves,  two  equations  between  the  two  unknown  quan- 
tities x^  and  jv',  and  can  readily  determine  them  by  algebraic 
methods.  If  we  find  only  one  pair  of  values  for  x^  and  y,  there 
is  but  one  point  of  intersection.  If  there  are  several  pairs  of 
values  for  x^  and  y^ ,  there  are  several  positions  of  P\  i.  e.  several 
points  of  intersection. 

We  will  illustrate  this  method  by  finding  the  point  of  inter- 
section of  the  curves  whose  equations  are 

2x— y—  \—o (i) 

and 

z^-y -z^^ »  •  (2) 

Let  P^  be  the  required  point.     Since  P^  is  a  point  of  the  first 
curve,  its  coordinates  satisfy  equation  (i). 
Therefore, 

2  a:'  -  7'  -  I  =  o (3) 

Similarly,  we  have 

3  ^'  -  J^'  -  3  =  o»     • (4) 

because  P^  is  a  point  of  the  locus  of  (2). 

Eliminating  between  equations  (3)  and  (4),  we  find 

x^  =  2,     y'  =$. 
The  curves,  therefore,  intersect  in  the  point  (2,  3). 
Let  the  student  verify  this  result  by  plotting  the  curves. 
We  will  next  find  the  intersections  of  the  loci  of 


and 

x-7y  + 

25  = 
25- 

0 

Proceeding 

as 

befo 

re,  we  have 

two 

equations 

to 

determi 

^ne  x^ 

and  y  :  — 

x'  -  jy'  + 
x'^+y"-- 

25- 

=  25 

-  0 

(t) 

, 

.(2) 

LOCUS    OF    AN    EQUATION.  2$ 

Eliminating  between  equations  (i)  and  (2),  we  find  that  either 

x'  =  ^  and  y'  =  4, 
°^  :»:'=-4andy  :=3. 

The  curves,  therefore,  intersect  in  the  two  points  (3,  4)  and 
( -  4,  3)  ;  as  will  be  seen  by  plotting  the  curves. 

It  will  often  happen  that  some  of  the  values  of  x'  and  v' 
are  imaginary.  The  geometrical  explanation  of  this  fact  may  be 
best  illustrated  by  an  example. 

The  points  of  intersection  of  the  loci  of 

X  +  y  z=  4     and     x'^  -\-  y'^  =  4 
will  be  found  to  have  for  their  coordinates 

(2  +  a/—  2,  2  —  V—  2)      and      (2  —  V—  2,  2  +  V—  2) 

Since  there  are  no  real  values  of  x'  and  7',  we  infer  that  there 
are  no  real  points  of  intersection  of  the  curves.  This  is  readily 
verified  by  constructing  the  curves,  which  will  be  found  to  be  a 
straight  line  and  a  circle  with  no  points  in  common.  Now  a 
straight  line  may  cut  a  circle  in  two  points,  and  hence  we  always 
find  two  sets  of  values  for  x'  and  y'  in  solving  such  an  exam- 
ple. These  values  may  give  two  real  and  different  points,  as 
in  the  second  example  of  this  section  ;  two  real  but  coincident 
points,  as  is  the  case  when  the  line  is  tangent  to  the  circle  ; 
two  imaginary  points,  when  the  line  falls  wholly  without  the 
circumference.* 

Note.  In  practice  it  is  not  customary  to  substitute  x'  andy 
for  X  and  y  in  the  equations,  but  to  eliminate  directly  between 
the  equations  of  the  curves,  and  thus  find  the  values  of  x 
and  y  which  will  satisfy  both  equations,  ^=^  i.  e.  the  coordinalts 
of  the  points  where  the  curves  meet.     It  is  safer  for  the  begin- 

*  It  is  important  for  the  student  to  understand  that  an  "imaginary  point  "  is  not  a 
point  at  all.  By  an  imaginary  point  we  mean  nothing  more  nor  less  than  a  pair  of 
imaginary  values  of  ^  and  j']  and  such  a  pair  of  values  is  not  represented  by  any  p^int 
whatever. 


24 


ANALYTIC    CxEOMETRY. 


ner,  however,  to  make  the  substitution;  for  he  must  always 
be  careful  to  remember  that  x  and  y  in  every  equation  are 
variables^  and  represent^  in  turn,  the  coordinates  of  each  and  every 
poifit  of  the  locus  of  that  equation. 


EXAMPLES. 

Find  the  points  of  intersection  of  the  curves  represented  by 
the  following  equations: — 


(i)    2X  — 2y—  \\— o,     2X— y  — ']=o. 

^"{2)    x^+y''  =  2S,     3^  +  47  =  25. 
(3)    •^■^+/  =  25,     xy  =  2. 


Ans.  (1,-4). 
Ans.  (3,  4). 


Ans.  < 


/A/29    +     -\/2I  A/29    —    a/2I\ 

(A/29    —     A/2I  A/29    +     a/2I\ 

2  '  2  / 

/  A/29    +    V2I  A/29    —    a/2i\ 

\~       ~2      '  r     / 

(—  a/29  —  A/21  A/29  4-  a/2i\ 

~2  '    ~  2  / 

(4)  y^  =  2  X,     X  —  2.  Ans.  (2,  2)  and  (2,  —  2). 

(5)  •^'-  5   ^+^  +  3    =    O.        ^^+/-5   A--37  +  6    rr   o. 

-^^^•f- 1  (3. 3),  (2,3),  (4,  0,  (i,  i)| 


W  :;  +  -^  =  i.    1  +  ^  =  ^' 


^     ^ 


^    ^ 


a  b  a  b 


/at?  a  0   \ 

'  \a  -^  b'     a  -\-  b) 


LOCUS    OF    AN    EQUATION.  25 

-^7)    The  equations  of  the  sides  of  a  triangle  are 

jv  4-  2  J'  -  5  z=  o, 

2x  +y  -  y  =  o, 
y-x-  1  =  o; 
find  the  vertices  of  the  triangle,  and  the  lengths  of  its  sides. 

(  V5'      V2'       V5      -> 
(8)    The  equations  of  a  circle  and  a  chord  of  that  circle  are 

x^  ^y"^  =  100  and  x  -}- y  =  14, 

respectively ;  the  centre  of  the  circle  is  the  origin  :  find  the  dis- 
tance of  the  chord  from  the  centre  of  the  circle. 

Ans.  7  ^"o 


mi  ^^^ 

U.  ot  C. 


CHAPTER   III. 


THE    STRAIGHT   LINE. 


21.  We  have  learned  in  the  preceding  chapter  that  any  equa- 
tion connecting  x  and  y  has  some  geometric  locus.  Conversely, 
it  is  true  that  any  curve  which  is  generated  by  a  point  moving 
according  to  some  law  has  for  its  equation  the  algebraic  expres- 
sion of  that  law. 

We  have  seen  that  the  loci  of  many  equations  are  straight 
lines.  Let  us  examine  the  distinctive  properties  of  straight 
Hues  and  express  them  in  an  equation,  —  i.  e.  find,  in  a  general 
form,  an  equation  which  will  always  represent  a  straight  line, 
but  which  may  in  turn  represent  all  straight  lines. 

22.  We  will  first  find  the  equation  of  the  straight  line  which 

cuts  O  Xiit  A  and  O  Y 
at  B.  By  changing  the 
positions  of  A  and  B, 
this  line  may  be  made 
to  take  any  desired  po- 
sition in  the  plane. 

The    lines    O  A    and 
OB  are  the  intercepts 
~x    of   AB,     and    will    be 
called  a  and  b,  respec- 
tively. 

Let  P  represent  any 
point  on  A  B,  and  draw  its  coordinates  O  M  —  x,  M  P  =  y. 


THE    STRAIGHT    LINE.  27 

Whatever  the  position  of  P,  MP  is  parallel  to  O  V,  and  the 
triangles  OP  A  and  MP  A  are  similar.     We  have,  therefore, 


MP      OB  _  b_ 
Wa  ^  0A~  a 


(0 


But 
and,  by  §  2, 


MP  =y, 


MA  =  M0+  OA 
=  OA- O  M 

—  a  —  X 

Substituting  these  values  in  (i),  we  have 

y     J 

a  —  X      a* 
or 

a  y  —  a  b  —  b  X ••••  (2) 

Equation  (2)  may  be  written  as  follows : 
b  X     ay       a  b 

a  b     a  b  ~  a  b 

or 

^-Hi  =  r (.) 

a     b  ^  ' 

The  last  form  of  the  equation  is  the  one  commonly  used  ;  it 
is  called  the  equation  of  the  straight  line,  in  terms  of  its 
intercepts. 

Note.  The  student  should  study  equation  (i)  carefully,  and 
convince  himself  that  it  is  true,  wherever  P  may  lie  on  the  line 
A  B, —  extended  indefinitely  in  both  directions,  —  but  cannot 
be  true  if  /^  moves  off  of  thai:  line.  If,  in  the  first  case,  P  and  B 
lie  on  the  same  side  of  O  A",  M P  2ir\d  O  B  have  like  directions 
and  algebraic  signs ;  the  Same  is  true  of  MA  and  O  A ; 
and,  therefore,  the  ratios  in  (i)  have  like  signs,  as  well  as 
*he  same  numerical  value.  If  P  and  B  lie  on  opposite  sides 
of  OX,  MP  and  O  B  have  opposite  directions  and  algebraic 


28  ANALYTIC    GEOMETRY. 

signs ;  the  same  is  true  oi  MA  and  O  A  ;  and,  again,  the  ratios 
of  (i)  have  like  signs,  together  with  the  same  numerical  value. 
When  -Pdoes  not  lie  on  A  B,  the  triangles  O B  A  and  M FA 
are  no  longer  similar,  and  the  ratios  of  (i)  are  unequal  numeri- 
cally. 

We  see,  then,  that  motion  of  F  along  the  straight  line  A  B 
is  consistent  with  equation  (i),  but  that  ^P  is  confined  to  posi- 
tions on  that  line.  Equation  (i),  therefore,  expresses  the  dis- 
tinctive properties  of  the  straight  line  ;  and  its  modified  forms, 
(2)  and  (3),  express  the  invariable  relation  which  exists  between 
the  coordinates  of  ajiy  point  of  A  B.  It  is,  then,  the  equation 
of  A  B,  according  to  the  definition  of  the  equation  of  a  curve, 
given  in  §  14. 

EXAMPLE. 

Find  the  equation  of  the  straight  line  which  makes  intercepts 
2  and  3  on  6^  Jf  and  O  F  respectively. 

Here 

a  —  2  and  6  =  3; 

therefore,  the  required  equation  is 

X     y 

-  +  -  =  I, 
2      3 
or 

^x-{-2y  —  6  =  0.     ^ 

23.  In  this  connection  it  is  important  to  notice  the  distinc- 
tions between  absolute  constants,  arbitrary  constants,  and 
variables. 

An  absolute  coiistarit  has  a  fixed  value,  the  same  wherever  that 
constant  appears.  3,  —  5,  \,  V2,  it  (the  ratio  of  the  circumfer- 
ence to  diameter),  etc.,  are  absolute  constants. 

An  arbitrary  cojistant  is  a  quantity  which,  though  fixed  in  any 
particular  problem,  has  different  values  in  different  problems 
of  the  same  class;  r,  the  circumference  of  a  circle,  and  r,  its 
radius,  are  examples  of  arbitrary  constants. 


THE    STRAIGHT    LINE. 


29 


A  variable  is  a  quantity  which  may  have  an  indefinite  number 
of  values  in  any  one  connection.  The  length  of  a  chord  of  a 
circle  is  a  variable,  because  in  the  same  circle  it  may  have  an 
indefinite  number  of  values. 

In  the  equation  of  the  straight  line  found  in  §  22,  we  have 
examples  of  each  of  these  three  kinds  of  quantities  :  i  is  an 
absolute  constant ;  a  and  b  are  arbitrary  constants,  fixed  when 
the  equation  represents  any  particular  line,  but  capable  of  other 
values  as  the  equation  represents  other  lines ;  x  and  y  are 
variables,  for  when  the  equation  represents  any  particular  line 
(i.  e.  with  each  pair  of  values  of  a  and  b),  these  quantities  are 
susceptible  of  innumerable  values,  as  they  represent,  in  turn,  the 
coordinates  of  all  points  on  the  line. 

24.  While  the  equation  of  the  straight  line  found  in  §  22  may 
represent  any  straight  line*  by  giving  appropriate  values  to  a  and 
b^  we  must  often  consider  straight  lines  which  are  not  known  to 
us  by  their  intercepts,  but  by  means  of  other  properties  of  the 
lines.  In  such  cases,  it  is  convenient  to  have  the  equations  of 
the  lines  in  terms  of  the  determining  quantities  which  are  given. 
We  shall,  therefore,  find  several  other  forms  of  the  equation  of 
a  straight  line. 

25.  A  straight  line 
is  determined  if  we 
know  where  it  cuts  O  V, 
and  also  its  di7'ection. 
The  first  of  these  ele- 
ments is  given  by  the 
intercept  on  O  V,  b; 
the  second  is  conven- 
iently expressed  by  the 
angle  which  the  line 
makes  with  (9^,  which 
we   will  call  y.      The 

*  Provided,  merely,  that,  tlie  line  does  not  pass  through  the  origin. 


y 

/4 

P 
R 

/y 

B 

T 

/' 

0                       P 

A                            X 

30  ANALYTIC    GEOMETRY. 

tangent  of  y,  which  is  called  the  slope  of  the  line,  we  will  indi- 
cate by  X. 

In  the  figure,  A  B  represents  any  line  ;  O  B  —  b;  and  the 
angle  X  A  Q  —  y.  The  angle  y  is  always  to  be  measured  from 
O  X  to  A  Q^  and  we  usually  measure  it  in  the  positive  direc- 
tion. P  represents  any  point  on  the  line,  and  its  coordinates, 
O  M  and  M  F,  are  x  and  y,  respectively.  Through  B  draw 
B  T  parallel  to  O  X,  and  in  the  positive  direction,  and  let  this 
line  and  MP  (or  their  extensions)  meet  in  P. 

From  the  figure  and  §  2,  we  have 

y  =  M  P=  MP  +  P  P 

=  O  B  +  P  P (i) 

But 

O  B  =  b, (2) 

and 

TB  Q^  XA  (2  =  y 

The  triangle  B  A'  7^  is  a  triangle  of  reference  for  y. 
From  this  triangle  we  have 
P  P 


or 

Since 
we  have 


^^  =  tany  =  A, 


P  P=  B  P  x\. 
BP  ^  OM=  X. 


PP=\x (3) 

Substituting  in  equation  (i)  the  values  found  in  (2)  and  (3), 
we  have 

y  =  b  -\-  Xx, 
or 

y  =  Xx  +  b, 

the  usual  form  of  the  equation  of  the  straight  line  in  terms  of  its 
intercept  and  slope. 


THE    STRAIGHT   LINE. 


31 


This  is  evidently  the  equation  we  seek  ;  for  it  expresses,  in 
terms  of  the  given  arbitrary  constants,  a  relation  [equation  (i)] 
which  exists  between  the  coordinates  of  any  point  of  A  jB,  but 
of  no  other  points  in  the  plane. 

Remarks. — If  y  is  in  the  ist  or  3d  quadrant,  A  (  =  tan  y)  is 
positive  ;  in  this  case  the  figure  shows  that  ^  P  and  B  i?  have 
like  signs. 

If  y  is  in  the  2d  or  4th  quadrant,  A  is  negative  ;  in  this  cas-j  it 
IS  easily  seen  that  R  P  and  B  R  have  unHke  signs. 

If  y  =  o  (i.  e.  if  the  line  is  parallel  \.o  O  X),\  —  o;  the  equa- 
tion, therefore,  reduces  to  the  form 

which  we  have  already  seen  must  represent  a  line  parallel  to 
OX{v.%  12). 

\i  b  —  o,  the  equation  becomes 

y  ■=z\x, 
which  is,  therefore,  the  equation  of  a  straight  line  through  the 
origin  ;  this  result  agrees  with  the  principle  stated  in  §  19. 

EXAMPLES, 
(i.)    Find  the  equation  of  the  line  which  cuts  (9  K  at  a  dis- 
tance of  two  units  above  the  origin,  and  makes  an  angle  of  45° 
with  OX. 

Here  A  =  tan  45°  =  i  and  b  =  2  ;  the  required  equation  is, 
therefore, 

7  =  jc  +  2. 

This  may  be  verified  by  plotting  the  locus. 

(2.)  Find  the  equations  of  the  lines  which  cut  O  Y  three 
units  below  the  origin,  and  make  angles  with  OX  of  135°,  30°, 
120°,  210°,  330°,  respectively. 

(3.)  Find  the  equations  of  a  pair  of  parallel  lines  making  an 
angle  of  150°  with  OX,  one  passing  through  the  origin,  the 
other  cutting  (9  Fat  a  distance  of  five  units  below  O. 

(4.)   x^"  +y-^  =  25  is  the  equation  of  a  circle  with  the  origin  as 


32  ANALYTIC  GEOMETRY. 

centre.  Find  the  coordinates  of  the  extremities  of  the  diameter 
which  has  a  slope  |. 

Ans.  (4,  3)  and  (-4,-3). 

26.  If  a  straight  line  passes  through  a  given  point  and  has  a 
given  direction  (indicated  by  its  slope),  it  is  determined  in  posi- 
tion. The  fixed  point  we  will  call  F^  or  {x^,  y^,  and  the  slope 
of  the  line,  A. 

We  will  deduce  the  equation  of  the  line  in  terms  of  these 
arbitrary  constants,  using  a  method  which  will  be  of  great  service 
hereafter. 

The  equation  of  §  25, 

y  ^\x  +  b, (i) 

will  represent  the  required  line,  if  we  can  find  such  a  value  for  b 
that  the  line  with  that  intercept  and  the  slope  A  shall  pass 
through  Fy 

This  we  can  easily  do;  for,  that  the  locus  of  (i)  may  pass 
through  F^,  it  is  necessary  and  sufficient  that  the  coordinates  of 
F^  shall  satisfy  (i),  when  substituted  therein  for  x  and  7.  Mak- 
ing this  substitution,  we  have 

y^  =  Xx^+b, (2) 

which  may  be  called  the  equation  of  condition,  that  the  locus  of  (i) 
shall  pass  through  F^,  or  that  F^  may  lie  on  the  locus  of  (1). 

Now,  in  equation  (2),  x^,  y^,  and  A  are  the  arbitrary  constants 
which  determine  the  position  of  the  line  whose  equation  we  seek. 
We  can,  therefore,  find  b  in  terms  of  these  quantities  by  solving 
equation  (2).     This  gives 

b  ^y^-\x^', 
substituting  this  value  of  <^  in  (i)  we  have 
y  =  \x  ^{y^-\  x^), 

which  may  be  written 

y-y,  =  \  (x-x,). 

This  must  be  the  equation  sought,  since  it  expresses  the  inva- 


THE    STRAIGHT    LINE.  33 

riable  relation  between  the  coordinates  of  every  point  of  the  Hne 
(and  of  no  other  points)  in  terms  of  the  required  arbitrary 
constants. 

The  student  should  verify  this  equation  by  deducing  it  from  a 
figure,  without  using  the  equations  of  the  straight  line  already 
found.     The  method  to  be  used  is  analogous  to  that  of  §  25. 

EXAMPLES, 
(i.)    What  is  the  equation  of  the  straight  hne  which  passes 
through  the  point  (2,  3)  and  has  an  inclination  of  45°  .? 

(  or  X  —  y  +  I  =0. 
(2.)    Find    the    equations    of   the   straight    lines  w^hich   pass 
through  (2,  —  4)  and  make  angles  of  60°  with  O  X  and  30°  with 
O  K,  respectively. 

j  +  4  =    V3  (a^_  2), 
J  +  4  =  -  V3  (^  -  2). 
(3.)    Find   the  equation    of   the   straight    line    which   passes 
through  the  point  common  to  the  loci  of 

2  X  —  y  —  \  —  o  and  t^  x  —  y  —  t^  =0, 

and  makes  an  angle  135°  with  OX. 

Ans.    x  -{-  y  -\-  1  =  o. 

(4.)  Find  the  equations  of  a  pair  of  parallel  lines  making  an 
angle  120°  with  OX,  and  passing  through  the  points  of  inter- 
section of  the  loci  of 

y  X  -\-  y  =  ^o  and  x^  +  y^  =  100. 

y+V3x-S-6V^  =  o, 


Ans. 


Ans. 


J  +  V3  -^  +  6  -  8  V3  =  o. 


27.  A  straight  line  is  also  determined  when  it  passes  through 
two  given  points.  Here  the  arbitrary  constants  are  the  coordi- 
nates of  the  given  points,  and  we  can  find  the  equation  of  the 


34  ANALYTIC    GEOMETRY. 

line  in  terms  of  these  quantities  from  the  equation  of  the  straight 
line  found  in  the  last  section. 

The  equation 

y  -}'!  =  ^(^  -^i) (0 

represents  a  straight  line  which  passes  through  one  given  point 
/'j,  and  has  a  slope  A.  By  giving  to  this  line  the  proper  direc- 
tion or  slope,  we  can  make  it  pass  through  the  second  given 
point  P.,.     Let  us  tind  the  appropriate  value  of  A. 

The  equation  of  condition  which  must  be  fulfilled  in  order 
that  the  locus  of  (i)  may  pass  through  jPg  ^s 

y2  -  yi  =  ^  (^2  -  ^i)  ', o  ....  (2) 

i.e.  the  coordinates  of  P.^  must  satisfy  equation  (i).  In  equa- 
tion (2),  A  is  the  only  unknown  quantity  ;  and  solving  for  A.,  we 
find  _ 

^  =  ^^-f (3) 

2         1 

Therefore  if  A  in  equation  (i)  has  this  value,  the  locus  of  (i) 
will  be  a  straight  line  through  /^^  and  P^.  Substituting  this  value 
of  A  in  (i),  we  have 

>'-^i  =  ^^(^-^.). (4) 

which  must  be  the  required  equation. 

This  equation  is  commonly  written  in  the  more  symmetrical 
form 


^2  -  yx     ^2 


■X 


(5) 


but  the  first  form  is  often  useful. 

Another  convenient  form  of  this  equation,  obtained  by  clear- 
ing of  fractions  and  reducing,  is 

O'l  -  72)  "^  -  (-^1  -Xo)y-\-x,y,-x^y,^o..,  (6) 

The  student  should  now  deduce  equation  (5)  from  a  figure, 
without  assuming  any  other  form  of  the  equation  of  the  straight 
line. 


THE    STRAIGHT    LINE.  35 

EXAMPLES, 
(i.)    Find  the  equation  of  tlie  straight  line  which  connects 
the  points  (4,  2)  and  (—3,  9). 

We  may  choose  either  point  as  P^,  and  the  other  will  be  P^. 
Let 

x^  =  4,     }\  =  2,     ^2  =  -  3,    j^2  =  9- 

Substituting  these  values  in  the  equation  just  found,  we  have 

J  -  2   ^    X   -  a, 

9-2        -3-4' 
or 

^  -  2   _  ^  -  4 

7       "     -7   ' 
which  may  be  written 

X  ^  y  —  d  =  o, 

the  required  equation  in  its  simplest  form. 

(2.)  Find  the  equations  of  the  sides  of  the  triangle  which  has 
for  its  vertices  the  points  (2,  i),  (3,  —  2),  (—  4,  —  i). 

^3^+7  -7  =  0, 
Ans.  <  X  -^  J y-\-  II  =0, 

C  X  -  sy  +  T-  =0. 

(3.)  In  the  same  triangle,  find  the  equations  of  the  straight 
lines  which  connect  the  vertices  with  the  middle  points  of  the 
opposite  sides. 

^  X  —      y  —  1  =  o. 
Ans.  \  X  +    2y  -\-  I  =  o. 
C  X  -  i2,y  -  9  =  o. 
(4.)    Find  the   equations  of  the  diagonals   of  the  rectangle 
formed  by  the  lines  whose  equations  are 

X  =  a,     X  =  a',    y  —  b,    y  =  b'. 

C  (^  _  l,f)  X  -  {a  -  a')  y  +  ab'  -  a'  b  =  o, 
'  \  (^b  -  b')  X  +  {a  -  a')  y  —  a  b  +  a'  b'  =  o. 

(5.)    Find  the  equation  of  the  straight  Hne  which  joins  the 


36  ANALYTIC    GEOMETRY. 

points  of  intersection  of  the  two  pairs  of  straight  lines  which 
have  the  equations 

2'^  +  3J'  —  4<^  =  o)       ^     X  -{-  6y  —  y  a  =  o 
^  ano 


2  X  +    y  —    <^=o)  (^^x  —  2y-\-2a  =  o. 

A?is.    4Ar  +  4j_5^  =  o. 
(6.)    Find  the  equation  of  the  straight  Hne  which  joins  the 
origin  to  the  intersection  of  the  loci  of 

Ax  +  By  -f  C  =  o  and  A'x  +  B'y  +  O  =  o. 

Ans.  {AC  -  A'C)  x  +  (BC  -  B'C)y  =  o. 

(7.)    Find  the  equations  of  the  diagonals  of  the  quadrilateral 
which  has  for  the  equations  of  its  sides 

3  ^  +  4  J  ._  13  ^  o,         X  -   6y  +    3  =  0, 
2^  +  3/  +6  =  0,      3:^+117  +  22=0. 

x  —      y  —   2  =  Oy 


A^jzs 

■  5;^+  147+  15  =  o, 

(8.)    Find  the  equations  of  the  medial  lines  of  the  triangle 
whose  vertices  are  the  points  B^,  Br,,  B^. 
Ans.    One  of  the  lines  has  the  equation 

the  other  equations  may  be  formed  from  this  by  advancing  the 
suffixes. 

(9)  Find  the  equations  of  the  same  lines  when  B^  is  the  ori- 
gin, and  B^  B^  is  the  axis  of  x. 

(2y^x+  {x^-  2  x^)  y-Xr,y,  =  o, 

Afis.  <     y^  X  +  {2  x^  —  x^)  y  —  x^ y^^  =  o, 

i     y^x-  {    x,+  ^2)  y  =0. 

(10.)  Confirm  the  results  of  Ex.  (9)  by  means  of  the  equa- 
tions found  in  Ex.  (8). 

(11.)  Prove  that  the  medial  lines  of  a  triangle  meet  in  a 
common  point. 


THE   STRAIGHT   LINE. 


37 


(i2.)  Prove  that  the  diagonals  of  a  parallelogram  bisect  each 
other. 

Suggestion.  Take  one  side  of  the  parallelogram  for  the  axis 
of  X,  and  one  extremity  of  that  side  as  the  origin. 

(13.)  Prove  that  the  lines  which  join  the  middle  points  of  the 
opposite  sides  of  a  trapezoid  bisect  each  other. 

(14.)  If  jE  and  jFare  the  middle  points  of  the  opposite  sides, 
A  Z>,  B  C,  oi  a  parallelogram  A  B  C  Z>,  the  straight  lines  B  £, 
D  F,  trisect  the  diagonal  A  C,  Prove  this  theorem  (v.  §§  10, 
II,  20,  27). 

(15.)   Prove  that  (2,  3),  (  —4,  —  i),  (8,  7)  lie  on  one  straight  line. 

28.  A  straight  line  is  determined  in  position  if  we  know  the 
length  of  a  perpendicular  drawn  to  it  from  the  origin,  and  the 
direction  of  this  perpendicular  (represented  by  the  angle  which 
it  makes  with  OX). 

We  will  next  find  the  equation  of  the  straight  line  in  terms  of 
these  arbitrary  constants. 

Let  A  B  repre- 
sent any  line  in 
the  plane,  O  Q 
the  perpendicular 
upon  it  from  O, 
and  XO  Q  the 
angle  which  de- 
termines the  di- 
rection of  O  Q. 
O  Q  IS  measured 
from  O ;  and  the 
angle  is  measured 
from  O  X  \c^  O  Q.  We  will  call  the  angle  a,  and  the  perpendic- 
ular/; it  snW  be  seen  that/  will  always  be  positive,  for  the 
positive  diniction  of  the  terminal  line  of  an  angle  is  from  the 
vertex  alon^  i^  boundary  of  the  angle. 


38  ANALYTIC    GEOMETRY. 

Let   P  be   any   point    of   A  B,   with  coordinates   x  =  O  M, 
y  —  3f  P.      From  M  draw   perpendiculars    to    6^  ^  and   A  B 
meeting  tliem  in  R  and  S  respectively. 
From  the  figure  and  §  3  we  have 

QO V= QOX^ XO Y 
= XO V- XOQ 

o 
=  90     —  a. 

If  we  always  take  7"  and  ^on  J/^  and  MP  respectively,  so 
that  J/ 7" shall  have  the  same  direction  as  its  parallel  O  Q,  and 
M  [/  3.S  O  V,  we  may  write 

TMU  =  QO  V=go°  -  a. 

Now,  O  R  M  is  always  a  triangle  of  reference  for  the  angle 
QO  X,  or  (—a);  and  MSP  is  a  triangle  of  reference  for  the 
angle  T  M  U,  or  (90°  —  a). 

From  the  figure  we  have,  by  §  2, 

p  =  OQ=  0R  +  RQ 

=  OR  +  MS (i) 

OR  ,       . 

Q-^=  cos  (-a)  =  cos  a; 


Now 


therefore, 
Again, 

therefore, 


OR  =  OMx  cos  a 

—  X  cos  a (2) 

j^  =  cos  (90    -  a)  =  sm  a ; 


MS  =  MP  X  sin  a 

=  J  sin  a (3) 

Substituting  in  equation  (i)  the  values  found  in  (2)  and  (3), 
we  have 

p  =  X  cos  a  4-  7  sin  a, 


THE    STRAIGHT    LINE.  39 

or  X  COS  a  +  y  sin  a  =  /, 

the  usual  form  of  the  equation. 

Since  this  equation  expresses  the  necessary  relation  between 
the  coordinates  of  every  point  of  the  straight  line  AB  (and  of  no 
other  point),  it  is  the  equation  of  that  line. 

Let  the  student  obtain  this  form  of  the  equation  of  the  straight 
line  from  the  equation  in  terms  of  the  intercepts, 

X       y 

-  +  7  =  I. 

This  may  be  done  by  expressing  a  and  b  in  terms  of/  and  a. 

Note.  —  It  is  important  to  notice  that  the  equation  which  we 
have  deduced  in  this  section  will  remain  the  equation  of  A  B,  if 
we  consider  Q  O  the  positive  direction  of  the  perpendicular, 
instead  of  O  Q.  P'or  the  angle  a  is  measured  from  O  X  \.o  the 
positive  direction  of  the  perpendicular,  and  (90°  —  a)  is  measured 
from  the  positii'e  diredioJi  of  the  perpendicular  to  O  Y. 

Therefore,  by  reversing  the  direction  of  the  perpendicular,  we 
change  each  angle  by  180°,  and  only  change  the  algebraic  signs 
of  their  cosines.  Each  term  in  the  equation,  therefore,  has  its 
sign  changed,  and  the  equation  is  still  true  for  all  points  on  A  B, 
and  for  no  other  points. 

EXAMPLES. 

(i.)  Find  the  equation  of  the  straight  line  such  that  the  per- 
pendicular upon  it  from  the  origin  bisects  the  angle  X  O  V,  and 
equals  V^* 

Here 

/  =  vT  ^^^  ^  —  45°  > 

therefore 

cos  a  =  Vg  and  sin  a  =  -v/J. 
The  required  equation  is,  therefore, 

X  Vi  +y  Vh  =  Vh 
or  X  -\-y  =  I. 


40  ANALYTIC    GEOMETRY. 

(2.)  A  circle,  with  a  radius  4,  has  the  origin  for  its  centre. 
Find  the  equations  of  the  tangents  to  the  circle  at  the  extremi- 
ties of  the  diameter  which  makes  an  angle  120°  with  O  X. 

29.  We  have  found,  in  §§  22-28,  the  equation  of  a  straight 
line  in  different  forms  ;  and  we  have  in  several  cases  shown  how 
one  of  these  forms  is  merely  a  modification  of  others. 

It  is  noticeable  that  in  each  form  the  equation  is  of  the  Jirst 
degree.  We  will  now  examine  the  general  form  of  the  equation 
of  the  first  degree  between  x  and  j,  and  show  that  it  always 
expresses  a  law  governing  these  quantities  such  that  the  point 
{x,  y)  will  be  confined  to  motion  on  a  straight  line. 

The  general  form  of  the  equation  of  the  first  degree  may  be 
written  :  — 

Ax-\.By+C=o', (i) 

where  A  represents  the  sum  of  the  coefficients  of  all  the  terms 
which  contain  x ;  B,  of  all  which  contain  y  ;  and  C,  the  sum  of 
all  terms  which  contain  neither  x  nor  y. 
Xf  we  solve  equation  (i)  for_r,  we  have 

AC  ,  ^ 

which  is  in  the  form 

y=\x\b, (3) 


'  A 

where  \  —  —-—  and  b 


—  -^ 

-  Bl 


whic  h  is  known  to  be  the  equation  of  a  straight  line. 

Now,  whatever  the  values  of  A^  B,  and  C,  —  —  is  the  slope  of 

B 

8om*  line.  For  the  slope  of  a  line  is  the  tangent  of  the  angle 
whi(  h  the  line  makes  with  O  X ;  and  as  this  angle  changes  from 
0°  t)  180°,  its  tangent  passes  through  all  possible  values  from  00 


THE    STRAIGHT    LINE.  4I 

to  —  00.     There  is  some  angle,  therefore,  between  o°  and  i8o°, 
which  has  its  tangent  equal  to  —  — ,  and  a  straight  line  having 

this  inclination  has  -  -^  for  its  slope. 

C 

Again,  —  —  is  the  intercept  of  some  line  on  O  V;  for  a  line 

may  cut  O  F  anywhere,  and  so  have  any  intercept. 

If,  then,  we  form  the  equation  of  the  straight  line  which  has 

A  C 

the  slope  —  -^,  and  —  ^-  for  its  intercept  on  O  V,  the  equation 

must  be 


=(-i)-(-f) 


or 

A  X  -ir  ^  y  -\-  C  =  o. 

Therefore^  whatever  the  values  of  A,  B,  and  C,  equation  (i) 
has  for  its  locus  some  straight  line.  But  (i)  represents  in  turn 
all  equations  of  the  first  degree  between  x  and  y.  Therefore,  all 
such  equations  are  equations  of  straight  lines. 

From  the  preceding  discussion,  we  see  that  the  values  of  the 
slope  and  intercept  on    O  V  can  be  easily  obtained  from   the 
equation  of  the  line.     For,  if  this  equation  is  put  in  the  form 
A  X  -{-  By  +  C  =  o, 


we  have 


A   ,  C 


Note.  The  student  should  notice  that  the  principle  of  this 
section  will  be  proved  if  we  can  put  the  general  equation  in  any 
of  the  forms  which  we  know  to  represent  straight  lines. 

EXAMPLES, 
(i)    Put  the  general  equation  of  the  first  degree  in  the  form 
X      y 
a      o 


42  ANALYTIC    GEOMETRY. 

and  obtain  values  for  a  and  b  in  terms  of  the  coefficients  A,  B, 

and  C.  A  ^    J  ^ 

Ans.a  =  --,  b  =  -- 

(2)  Calculate  the  intercepts  and  slope  of  the  straight  line 
whose  equation  is 

2^  +  37  —  6  =  0. 
Here 

^  =  2,   ^  =  3,    C  =  -6, 
and  we  easily  find 

«  =  3,   b^2,   A  =  --. 
o 

(3)  Find  the  intercepts  and  slope  for  each  of  the  lines  which 
have  the  following  equations  :  — 

X  cos  o.  ■\-  y  sin  a  z=  p. 
30.  It  is  obvious  that  two  equations,  of  which  the  second 
can  be  obtained  from  the  first  by  muliiplication  by  a  constant 
factor,  will  have  precisely  the  same  locus,  since  any  pair  of 
coordinates  which  satisfies  one  of  these  equations  also  satisfies 
the  other.     Thus  the  two  equations, 

2^  —  3^—  6  =  0, 
6x  —  gy  —  18  =  0, 

both  represent  the  line  whose  intercepts  are  3  and  —  2. 

No  less  important,  though  perhaps  less  obvious,  is  the  fol- 
lowing fact 

//  iivo  equattofis  of  the  first  degree  have  the  same  locuSy  the  sec- 
ond can  be  obtained  frofn  the  first  by  7nultiplication  by  a  constant 
factor. 

For,  let 

Ax-^By-\-C  =  o, (i) 

A^x  ^  Bj  +  Cj  =  o, (2) 


THE    STRAIGHT    LINE.  43 

be  the  two  equations  of  the  first  degree  which  are  supposed  to 
have  the  same  locus.  This  locus,  as  we  know,  is  a  straight 
line  whose  slope  A  and  intercept  b  may  be  computed  from  the 
first  equation  by  the  formulae 

from  the  second  equation  by  the  formulae 

When  we  remember  that  the  two  equations  are  supposed  to  rep- 
resent one  and  the  same  line,  we  see  that  the  two  values  of  A 
just  obtained  must  be  equal;  and  the  same  is  true  of  the  two 
values  of  b.     Therefore 


B  B^'  B  B^* 


ABC 


(3) 


A 
Let  us  now  multiply  (i)  by  the  constant  -— ^.     This  gives 

A  B 

or,  when  we  replace  — ^  in  the  second  term  by  its  value  ^,  and 
^         A  B  * 

C 

in  the  third  term  by  its  value   ~  [v.  (3)  ], 


44  ANALYTIC    GEOMETRY. 

and  this  reduces  at  once  to  equation  (2).     Thus  we  see  that 

A 
(2)  may  be  obtained  by  multiplying  (i)  by  the  constant  -j- , 

31.    Let  us  reduce  the  general  equation  of  the  first  degree  to 
the  form  found  in  §  28. 

Whatever  the  locus  of 

A  X  +B  y  +C=o, (i) 

its  equation  may  be  expressed  in  the  form 

X  cos  a+jsina— /z=o; (2) 

for  every  straight  line  can  have  its  equation  written  in  any  of  the 
forms  found  in  §§  22-28. 

Regarding  (i)  and  (2)  as  equations  of  the  same  straight  line 
their  first  members  must  either  be  identical,  or  one  must  be  some 
multiple  of  the  other  (v.  §  30).      Let  the  unknown  multiplier, 
which  will  make  (i)  identical  with  (2),  be  represented  by  Ji. 

'^^^^"'  RAx-^RBy-\.RC=o, (3) 

and                                                       .  ,  ^ 

X  cos  a  +  J'  sm  a  —  /  =  o (4) 

are  identical.     Therefore 

A'  ^  :^  cos  a,    B  B  =  sin  a,  R  C  =  -/. 

We  have  from  Trigonometry 

cos'^  a  +  sin^  a  r=  I  ; 

substituting  the  values  of  cos  a  and  sin  a, 

and  solving  for  A^, 

^2 I 

-A'  +  B^' 

and 

R 


±  Va'  4-  B^ 


THE    STRAIGHT    LINE.  45 

Using  this  value  of  R,  we  hav^e 


cos  a  — 


Sin  a  = 


/  =  - 


c 


We  may  use  either  sign  with  the  radical,  but  must,  of  course, 
use  the  same  one  in  the  values  of  cos  a,  sin  a,  and  /.  We  have 
decided  to  take  /  positive ;   therefore,   we   will   in    each    case 

C 

choose  that  sign  with  the  radical  which  will  make  —      .  ^m 

^/a^  +  B 
positive.     This  must  be  the  opposite  sign  to  that  of  C. 

Note.  —  The  reason  why  we  may  choose  either  sign  with 
"s/ A^  +  B'  (i.  e.  have  two  multipliers,  R)^  is  found  in  the  note  to 
§  28.  It  is  there  shown  that  the  same  line  will  have  two  equa- 
tions in  the  form 

X  cos  a  -\-  y  sin  a  —  p, 

if  we  allow  either  algebraic  sign  with  //  and  that  the  values  Ol 
cos  a,  sin  a,  and/,  will  be  the  same  numerically,  but  have  oppo- 
site signs  in  the  two  equations. 

Now,  if  we  choose  that  sign  for  \/A^  +  B'^  which  makes  p 
positive,  we  get  certain  values  for  cos  a  and  sin  a ;  these  values 
will  only  have  their  signs  reversed  when  we  change  the  sign  of 
VA'"^  +  B'^,  so  that  the  second  value  of  a  differs  from  the  first  by 
180°,  as  was  shown  in  the  note  to  §  28. 

EXAMPLES. 

(i.)    Reduce  the  equation 

^x-sy-y =0 
to  the  form 

X  cos  a  +  J  sin  a  =/. 


46 

ANALYTIC    GEOMETRY. 

Here 

therefore 

^  =  4,  ^  =  -3,  C=  -7; 

/?-               '                      '              ^ 

+  V^2  +  ^2      y  j6  ^.  ^      5 

We  choose  the  positive  sign  with  the  radical,  b^ 

of  C  is  negative. 

^.-^7  =  7 
5          5         5 

is,  therefore, 

the  required  equation,  and 

4      .                3            7 

cos  Ci  =  -,    SUl  a  =  -",/=  i-. 

(2.)  Find  the  distance  from  the  origin  to  the  line  which  has 
the  equation 

2  ^3  .r  —  2  _y  +  7  =  o, 

and  the  angle  which  this  perpendicular  makes  with  O  X. 

Afis.  -  :  ii;o°. 
4        ^ 
(3.)    Find  the  length  of  the  perpendicular  from  the  origin  on 
the  line  whose  equation  is 

a  {x  —  a)  ■\-  b  {y  —  b)  =0, 

32.  In  the  preceding  sections  we  have  found  those  forms  of 
the  equation  of  the  straight  line  which  will  be  most  useful  here- 
after. We  have  also  shown  that  any  equation  of  the  first  degree 
between  x  and^  must  represent  a  straight  line. 

We  will  now  make  use  of  the  equations  which  we  have  found 
to  solve  several  problems  relating  to  straight  lines  and  points, 
the  results  of  which  are  convenient  for  reference. 

33.  Let  us  find  the  angle  between  two  straight  lines  given  by 

their  equations  :  — 


THE    STRAIGHT    LIX 


47 


P^rst,  suppose  the 
equations  of  the  Hues 
to  be  in  terms  of  inter- 
cept and  slope.  We 
may  write  the  equa- 
tions : 

y  =  Xx  +  b,  .  .  (i) 

y  =  A,  X  +  /?,  .  .  (2) 
Here,  A  =tan  ■; ,  and  Aj  = 
tan  y,,  7  and  y^  being 
the  angles  which  the 
lines  (i)  and  (2)  re- 
spectively make  with 
O  X,  as  represented  in 
the  figure.  Let  O  be  the  angle  which  the  line  (2)  makes  with 
the  line  (i)  ;  i.  e.  t>  is  measured  y)'6'w  (i)  to  (2).  This  is  the 
angle  we  seek. 

By  §  3»  7  +  r'^  =  y„ (3) 

is  is  evident  when  we  draw  through  T,  the  point  of  intersection 
of  the  lines,  a  line  parallel  to  O  X. 
From  (3),  we  have 

^  =  vi  -  y- 

Therefore, 


-        tan  71  -  tan  7 
tan  t9-  = '—   . 

I  +  tan  7^  tan  7 

Substituting  the  values  of  tan  y^  and  tan  7,  we  have 


tan  19^ 


A,  -A 
I  +  A,  A 


(4) 

This  gives  tan  (y  in  terms  of  known   quantities,  and  ^  can  be 
easily  found  from  its  tangent  by  Trigonometry. 


48  ANALYTIC    GEOMETRY. 

If  the  equations  of  the  lines  are 

Ax+By-rC=o, (i') 

A,x  +  B,y  +  C^  =  o, (2') 

we  have  only  to  find  the  values  of  X  and  Aj  in  terms  of  A,  B,  A^^ 
B^,  and  substitute  the  values  thus  found  in  (4). 
By  §  29, 


Therefore, 


1 


or 


X  =  - 

^andA,=_-. 

tan  ^  - 

A 
A 

-i-i) 

-(- 

U-i)' 

tan  i 

-ff 

\-A,B 

■  +  -5  ^i       

(5) 


Note.  —  It  should  be  carefully  remembered  that  ^  is  meas- 
ured from  (i)  to  (2),  and  so  will  be  positive  or  negative  accord- 
ing as  -/i  is  greater  or  less  than  y.  By  interchanging  the  lines 
(i)  and  (2),  therefore,  we  shall  change  the  sign  of  i^,  and  so 
change  the  sign  of  its  tangent.  The  latter  fact  is  also  evident 
from  the  values  of  tan  i>,  in  either  of  which  the  proposed  inter- 
change will  reverse  the  sign  of  the  fraction. 

Corollary  1.  If  the  lines  (i)  and  (2)  are  parallel,  the  angle 
i9-  is  0°,  or  180°,  and 

tan  Or  =.0, 

That  this  may  be  true,  we  must  have  the  numerators  of  the 
fractions  in  (4)  and  (5)  equal  to  o. 

The  conditions  for  parallel  lines  are,  therefore, 

Aj  —  A  =  o,  or  Aj  =  A ; 
or, 

A  B^-A,B  =  0,  or  AB,=  A^B. 


THE    STRAIGHT    LINE. 


49 


Corollary  2.     If  the  lines  are  mutually  perpendicular,  &  is 
90°,  or  270°,  and 

tan  19-  =  00 . 

Making  the  denominators  of  the  fractions  in  (4)  and  (5)  equal 
to  o,  we  have  for  the  conditions  of  perpendicular  lines 

I  +  X^kz=o,  ovX^  =  --; 

or, 

A  A^  +  B  B^  =  o,or  A  A^  =  -  B  B^. 

We  may  state  the  results  of  these  corollaries,  thus : 
Parallel  lines  have  the  same  slope. 

Lines  at  right  angles  to  each  other  must  have  the  slope  of  one 
equal  to  the  negative  of  the  reciprocal  of  the  slope  of  the  other. 

EXAMPLES. 

(i.)    Find  the  angle  which  the  line 

makes  with  the  line 

Here,  by  the  statement  of  the  question,  the  angle  is  to  be 
measured /r^;;?  the  last  line.     We  have,  then, 

X  =  4,     Ai  =  -  3, 
and 

I  +(-3)  4      11' 
therefore 

^  =  tan  - '  — 

If  we  use  the  other  formula  we  have 

^  =  4,     B  =  -  1,     A,=s,     A  =  i> 
and 

4  X  3  +  (-   I)  X  I       II 
which  agrees  with  the  previous  result. 


50  ANALYTIC    GEOMETRY. 

(2.)    Find  the  angle  which  the  line 

3-^+7  =  7 
makes  with  the  line 

2x^y  =  ^. 

Ans.   — . 
4 
(3.)   Find  the  angle  which  the  last  line  mentioned  in  Ex.  (2) 
makes  with 

3^+^  =  5. 

(4.)   Find  the  angles  of  the  quadrilateral  mentioned  in  §  27, 
Ex.  7. 

C  22  15  I  21   > 

Ans.  <  tan~'  —  — ,     tan"'  -^,     tan"' ,     tan~'  —  V 

C  21  16  3  535 

(5.)    Find  the  equation  of  the  straight  line  which  cuts  O  V 
four  units  below  the  origin,  and  is  parallel  to  the  line 

2x-  sy  +  4=0; 
find  the  equation  of  a  line  which  has  the  same  intercept  on  O  Y, 
and  is  perpendicular  to  the  line 

S      2 

'2Jt:  —  5^—  20  =  0, 


'-\ 


Ans. 

-  t^x  +  2y  +    5  =  0. 

(6.)  Recollecting  that  the  angle  between  two  lines  is  equal  to 
the  angle  between  the  perpendiculars  upon  those  lines  from  the 
origin,  find  an  expression  for  the  tangent  of  the  angle  which 
the  line 

X  cos  a.^+  y  sin  04  =/j 
makes  with  the  line 

X  cos  o'  +  y  sin  or  =  ^, 
(7.)  By  the  same  method,  deduce  the  formula  already  found 
for  the  tangent  of  the  angle  which  the  line 

A^x  +  B^y  +  C,  ^o 
makes  with  the  line 

Ax^By^C=o. 


THE    STRAIGHT    LINE.  5l 

(8.)    Find  the  angle  between  the  lines 

4^-3J-f5=o» 
and 

6x  +  Sy  +  ly  =  o. 

(9.)  Show  that  the  diagonals  of  a  square  are  perpendicular  to 
each  other. 

(10.)  Show  that  the  second  quadrilateral  mentioned  in  §  11, 
Ex.  10,  is  a  parallelogram. 

34.  It  often  happens  that  we  have  a  line 

Ax  +  By+Cr=o, (i) 

and  wish  to  find  the  equation  of  another  line  parallel  or  per- 
pendicular to  it.  The  following  considerations  are  very  useful 
in  such  cases. 

The  slope  of  any  line  parallel  to  (i)  is  —  (§   t,^,  Cor.  i). 

Accordingly  the  equation  of  such  a  line  may  Ije  written  (§  25) 

y  =  -^x  -h  ^ 

or 

Ax  +  By  -  dB  =z  o. 

7 he  equation  of  any  line  parallel  to  ^i)  may  be  so  written  that 
its  equation  differs  from  ( i )  only  in  the  value  of  the  constant 
term. 

Similarly  the  equation  of  any  line  perpendicular  to  (i)  may 
be  written  (§  33,  Cor.  2) 

B 

or 

B  X—  Ay  -V  b  A  —  o. 

The  equation  of  any  line  perpendicular  /^  ( i )  may  he  obtained  hv 
interchanging  the  coefficients  of  x  and  y  in  (  i )  and  changing  the 
sign  of  one  of  them^  and  (hen  suitably  alte*-ing  the  constant  term. 

These  two  rules  enable  us  to  write  down  the  equation  of  the 
desired  line   completely,  except   for  the  constant  term.      The 


52  ANALYTIC    GEOMETRY. 

fact  that  the  constant  term  is  not  determined  by  these  rules 
should  not  surprise  us,  for  the  mere  statement  that  the  desired 
line  is  to  be  parallel  or  perpendicular  to  (i)  does  not  deter- 
mine the  line  completely.  Some  further  statement  with  re- 
gard to  the  position  of  the  line  must  be  made  before  the  con- 
stant term  can  be  determined.  This  may  be  done,  for  instance, 
as  follows: 

Let  us  consider  the  problem:  To  find  the  equation  of  the 
line  which  passes  through  the  point  (x^,  }\)  and  is  parallel  to  (i). 

Since  this  line  is  to  be  parallel  to  (i),  its  equation  can,  by 
the  first  of  the  above  rules,  be  written  in  the  form 

Ax  +  By  —  k, (2) 

where,  for  convenience,  we  have  written  the  constant  term  k 
on  the  right-hand  side  of  the  equation.  The  question  which 
remains  to  be  settled  is:  What  is  the  value  of  the  constant  k  ? 
Since  the  line  (2)  is  to  pass  through  (a:,,  y^,  the  following 
equation  of  condition  must  be  satisfied: 
A  x^  -\-  B y^  —  k^ 
and  it  will  be  seen  that  this  equation  gives  the  value  of  the 
unknown  constant  k  in  terms  of  the  known  constants  A^  By 
a:,,  y^.  Substituting  this  value  of  k  in  (2),  we  find  as  the  equa- 
tion of  the  desired  line 

Ax  ■{■  By  =  A x^  +  By^. 

By  precisely  the  same  method  we  find  as  the  equation  of  the 
line  which  passes  through  the  point  (^j,  y^)  and  is  perpen- 
dicular to  (i) 

Bx  —  Ay  =  Bx^  —  Ay^, 

EXAMPLES, 
(i.)  Find  the  equations  of  two  straight  lines  through  (2,  — 1\ 
parallel  and  perpendicular,  respectively  to  the  line 
5x  +  3j+7  =  o. 


Ans.  \2  X  -y  -  1  =  o 
2y  +  3  =  o. 


THE    STRAIGHT    LINE.  53 

(2.)  Find  the  equations  of  two  lines  through  the  origin,  par- 
allel and  perpendicular,  respectively,  to  the  line 

3-^+5>'-4  =  o- 
(3.)   Find  the  equation  of  the  line  which  passes  through  the 
point  common  to  the  lines 

X  —  2y  —  a  =  o, 
and         X  -{-  ^y  —  2  a  ^=  o, 
and  is  parallel  to  the  line 

30:  +  47  =  o. 

Ans.  3^  +  4_y  —  5^  =  0. 
(4.)  Find  the  equations  of  the  perpendiculars  from  the  ver- 
tices to  the  opposite  sides  of  the  triangle  given  in  §  20,  Ex.  7. 

(x-\-y-4  =  o 

V  X 

(5.)  Through  the  vertices  of  this  triangle  lines  are  drawn 
parallel  to  the  opposite  sides ;  find  the  equations  of  the  sides  of 
the  triangle  thus  formed. 

^  X  —  y  —  2  =  o 
Ans.  \  X  +  2  y  -  d>  =0 
\2X-\-y  —  ^=0. 
(6.)  Find  the  condition  that  the  lines 
X  +  [a  +  b)  y  +  c  =  o, 
and  a  [x  +  ay)  +  b  (x  —  by)  -f  ^/  =  o 

KKiy  be  parallel;   that  tliey  may  be  mutually  perpendicular. 

Ans.   b  =  o,     b'  —  a''  =^  u 
(7.)   Choosing  the  axes  as  in  §  27,  Ex.  9,  prove  that  the  per- 
pendiculars erected  at  the  middle  points  of  the  sides  of  a  tri- 
angle meet  in  the  point  (-,  — — ^- — -). 

\  2  2  y^  ' 

(8.)  In  a  similar  manner,  show  that  the  perpendiculars  from 
the  vertices   to   the   opposite   sides   of  a  triangle  meet  in   the 

point  (^,,    fL^^). 


54 


ANALYTIC    GEOMETRY 


(9.)  Prove  that  the  common  intersection  of  the  medial  lines 
of  a  triangle,  the  intersection  of  the  perpendiculars  at  the  middle 
points  of  its  sides,  and  the  intersection  of  the  perpendiculars 
from  the  vertices  to  the  opposite  sides  lie  on  one  straight  line. 

(10.)  Show  that  the  first  of  these  points  is  one  of  the  points 
of  trisection  of  the  line  which  joins  the  other  two.  To  which  of 
these  points  is  it  nearer?     (v.  §  ji.) 

35.  Let  us  find  the  distance  of  a  given  point  P^from  a  given 
straight  line. 

Let  ^^  be  the 
given  line,  and  let  its 
equation  be 

X  cos  a  Ary  sin  a  =/>(l) 
Draw  a  line,    (2), 
through    /'p    parallel 
to  (i)  ;  and  from   O 
draw  a  perpendicular 
to  the  lines  (i)  and 
(2),  meeting  them  in 
Q    and     Q',    respec- 
tively,   ^/^i,  perpen- 
dicular to  A  B,  measures  the  required  distance  of  P^from  A  B. 
In  the  figure, 

O  Q  =p     and     X O  Q  =  a. 

We  have  chosen  p,  measured  from  O  to  Q,  positive ;  this  is, 
therefore,  the  positive  direction  of  lines  parallel  to  O  Q. 

Since  (i)  and  (2)  are  parallel  lines,  the  perpendiculars  upon 
them,  from  (9,  make  the  same  angle,  a,  with  OX;  the  equation 
of  (2)  is,  therefore, 

X  cos  a  +  jj^  sin  a  =  /', (2) 

where/'  =  O  Q'. 

From  the  figure  and  §  2,  we  have 

J^Pi=  QQ'=  O  Q'  -  OQ 

=/'-/ (3) 


THE    STRAIGHT    LINE.  55 

Now,  P^  is  a  point  of  the  line  (2)  ;  therefore  its  coordinates 
must  satisfy  equation  (2),  and  we  have 

x^  cos  a^-  y^  sin  a  =/'. 

Substituting  this  value  of/'  in  equation  (3),  we  have 

R  F^  —  x^  cos  a  +  j'^  sin  a  — /, (4) 

which  expresses  the  required  distance  in  terms  of  known  quan- 
tities. 

Note.  We  see  from  the  figure,  that  R  P^  is  positive  or  nega- 
tive according  as  P^  and  O  are  on  opposite  sides  oi  A  P  or  on 
the  same  side.  The  same  result  is  obtained  from  equation  (4). 
For,  in  the  first  case,/'  is  greater  than/,  and  both  are  positive, 
in  the  second,  when  P^^  is  nearer  A  B  than  O  is,/'  is  positive 
and  less  than/;  when  P^  is  farther  from  A  B  than  O  is,/'  will 
itself  be  negative,  according  to  the  principle  explained  in  §  28, 
Note. 

36.  We  may  solve  the  problem  of  the  last  section  when  the 
given  line  has  the  equation 

Ax^By^C^Q (i) 

by  reducing  equation  (i)  to  the  form 

X  cos  a  -\-  y  sin  a  =/, 
as  explained  in  §  31. 

The  value  of  the  required  distance  is,  therefore, 
A  B         _  (     C_ 

^^  VA-'  +  B'  "^  '^'  VA"  +  B'  ~  ^~  \/A^+~B' 
or 

Ax^  +  By.^C 

VA''  +  B' 

where  the  radical  has  the  opposite  sign  to  that  of  C  in  order  to 
make 


..(2) 


^[=-v^Jp°"'"" 


56  ANALYTIC    GEOMETRY. 

EXAMPLES, 
(i.)    How  far  is  the  point  (3,  -  2)  from  the  line 

3^-4J  +  2  =  o? 
The  formula  of  §  36  shows  that  the  required  distance  is 
3X3-4(-2)+2__ 

-  Vf+i-^r 

The  negative  sign  indicates  that  the  point  lies  on  the  same  side 
of  the  line  with  the  origin. 

(2.)    Calculate  the  distance  of  the  point  (7,  4)  from  the  line 

8  X  +  6  7  =  45. 

Ans.  3i  ;  and  the  line  lies  between  the  point  and  the  origin. 

(3.)    Find  the  distances  of  the  origin  from  the  lines 

X  cos  a  -\-  y  sin  a  =p, 

A  x  +  B  y  +  C  =  o. 

C 

(4.)    In  the  triangle  given  in  §  20,  Ex.  7,  find  the  lengths  of  the 
perpendiculars  from  the  vertices  to  the  opposite  sides. 
•   (5.)    Find  the  lengths  of  similar  lines  in  the  triangle  given  in 
§  9,  Ex  3,  and  show  by  your  results  that  the  origin  lies  within 
the  triangle. 

(6.)  We  may  obtain  the  distance  between  a  given  point  and 
a  given  line  by  finding,  first,  the  equation  of  the  perpendicular 
from  the  point  to  the  line  ;  then,  the  point  where  this  perpen- 
dicular meets  the  given  line  ;  and,  lastly,  the  distance  between 
this  point  and  the  given  point. 

By  this  method,  calculate  the  distance  of  the  point  (—  S,  3) 
from  the  line 

3^7-27  +  4  =  0. 

Confirm  your  result  by  applying  the  formula  of  §  36. 


THE    STRAIGHT    LINE. 


57 


(7.)  Using  the  same  method,  show  that  the  distance  between 
/\  and  the  Hne 

A  X  -{-  By  +  C •=.  o 
is 

A  x^  ■\-  B  y^^  C 

(8.)  If  the  given  point  lies  on  the  given  line,  its  distance  from 
the  line  is  nothing.  Show  that  the  formulas  of  §§  35,  36,  give 
the  same  result. 

(9.)  Two  parallel  lines  are  drawn  at  an  inclination  &  to  the 
axis  of  X :  one  through  the  point  {a^  b),  the  other  through  (a',  b'). 
Show  that  the  distance  between  these  lines  is 

{a'  -  a)  sin  {>  -  {b'  -  b)  cos  ^. 

37.  The  area  of  a  triangle  can  be  found  when  the  coordi- 
nates of  its  vertices  are  given.  For  we  have  only  to  calculate 
the  length  of  one  side, 
by  §  10,  and  the  perpen- 
dicular upon  this  side 
from  the  opposite  vertex, 
by  §  36.  The  half-prod- 
uct of  these  quantities  is 
the  measure  of  the  area. 
Let  F,,  P,,  P.,  be  the 
vertices  of  the  triangle. 
The  equation  of  the  line 
P,P.^{\.^.  of  the  indefi- 
nite line  determined  by 
P.^  and  P3)  is,  by  §  27, 

The  distance  of  P^  from  this  line  is,  by  §  2)^^ 

0'2  -  J'3)  -^1  -  (^2  -  ^3)  }\    +    ^2  ->^3  -  ^3  y-1 


(0 


p  p  - 


V(/2  -  i's)'    +    (^2  -  ^3)' 


(2) 


58  ANALYTIC    GEOMETRY. 

By  §  10,  we  have 


p,  p,  =  VO;  -y^'  +  {X,  -  x,y (3) 

If  A  represents  the  area  of  the  triangle, 

=  (J2  - y^^  ^1  -  (-^2  -  ^3)  yi  +  ^2^3  -  ^3  J2  •  •  •  •  (4) 

Equation  (4)  may  be  written 

2  ^  =  (^1  -  X,)  7,  +  (at,  -  ^3)  y^  +  {x^  -  x^  y^. 

This  form  is  evidently  symmetrical  with  respect  to  the  coor- 
dinates of  i",,  P^,  and  P.^,  as  should  be  the  case  ;  for  it  is,  of 
course,  immaterial  which  side  is  used  as  the  base  of  the  triangle, 
when  calculating  the  area. 

Note.  As  only  the  Icngi/i  of  P^  P^  is  needed  in  calculating 
the  area  of  the  triangle,  wc  need  only  notice  the  numerical  value 
of  our  result.  The  algebraic  sign  will  generally  differ  when  we 
take  different  sides  for  the  base. 

EXAMPLES. 

(i.)  Find  the  area  of  the  triangle  which  has  for  its  vertices  the 
points  (2,  3),  (-  I,  4)  (6,  -  5). 

Calling  the  points  P^,  P.^,  and  iPj,  respectively,  we  have 

2  ^  =  [2  -  (-  I)]  (-  5)  +  (-  I  -  6)  3  +  (6  -  2)  4 

rr     —    15    —    21    +    16    =    —    20. 

Therefore  the  area  is  10. 

The  unit  of  area  is,  of  course,  the  square  which  has  each  side 
equal  to  the  unit  of  length  in  terms  of  which  the  coordinates  of 
the  given  points  are  expressed. 

(2.)    What  is  the  area  of  the  triangle  given  in  §  20,  Ex.  7. 

(3.)    F'ind  the  area  of  the  quadrilateral  given  in  §  27,  Ex.  7. 

Ans.  y. 

(4.)  Obtain  the  formula  for  the  area  of  the  triangle  P,  P,,  P^ 
[§  37,  Figure]  by  subtracting  the  trapezoid  J/g  P.^  P.^  J/,  from  the 
sum  of  the  trapezoids  J/^  iR,  ^\M^  and  M^  P^  P.  M.^ 


THE    STRAIGHT    LINE.  59 

(5.)  Prove  that  the  three  straight  lines  joining  tiie  middle 
points  of  the  sides  of  a  triangle  divide  the  triangle  into  four 
equal  triangles. 

38.  In  all  our  work  hitherto,  we  have  used  the  rectangular 
system  of  coordinates.  It  is  often  more  convenient  to  denote  the 
position  of  a  point  by  means  of  its  di)'ection  and  distance  from  a 
fixed  point  in  the  plane,  than  by  means  of  its  distances  from  two 
fixed  lines. 

If  O  X\x\  the  figure  is  a  fixed  line, 
and  O  a  fixed  point  of  that  line,  the 
position  of  P  is  completely  given  by 
the  angle  X  O  F  =  (\>,  and  the  dis- 
tance O  P  =  r. 

The  two  elements,  or  coordinates^ 
which  determine  the  position  of  the 

point  P  are,  therefore,  r,  called  the  radius  vector,  and  (/>,  called 
the  vectorial  angle ;  ^  is  measured  from  the  fixed  straight  line, 
called  the  initial  line,  and  r  from  O,  called  the  pole. 

This  system  of  coordinates  is  called /^'/^r. 

It  will  be  readily  seen  that  the  simple  equation 

r  —  a 

has  for  its  locus  the  circumference  of  a  circle  with  (9,  the  pole, 
for  its  centre,  and  a  for  its  radius.  This  example  is  sufficient  to 
show  the  student  that  certain  problems  may  be  conveniently 
treated  by  this  system  of  coordinates. 

39.  We  will  find  the  equation  of  the  straight  line,  using  the 
polar  system. 

In  the  figure  on  the  next  page,  O  is  the  pole,  O  X  the  initial 
line,  and  AB  any  straight  line  in  the  plane. 

The  position  of  the  line  is  determined  by  /,  equal  to  O  Q, 
the  perpendicular  from  the  pole  to  the  line,  and  a,  equal  to 
X  O  Q,  the  angle  which  /  makes  with  the  initial  line. 


6o 


ANALYTIC   GEOMETRY. 


Let  P  be  any  point  on  A  B,  with  coordinates 

r  =  OF,     (f>  =  XOP. 
Now  from  the  figure  and  §  3 
we  may  write 

QOP= QOX+ XOP 
= XOP- XOQ 

=  ct>-a (i) 

Again  ;  the  triangle  Q  O  P 
must  be  a  triangle  of  reference 
for  the  angle  Q  O  P,  and  there- 
fore we  may  write 


OQ 
OP 


=  cos    QOP  =  cos  {4,  -  a) 


(2) 


Substituting  in  (2)  the  values  oi  O  Q  and  OP,  and  clearing 
of  fractions,  we  have 

r  cos  ((})  —  a)  =  p, 

which  must  be  the  required  equation,  since  it  expressejs  a  neces- 
sary relation  between  r  and  <^,  the  coordinates  of  any  point  on 
A  B  (and  of  no  other  points),  in  terms  of  the  chosen  arbitrary 
constants,/  and  a. 


\ 


CHAPTER  IV. 


TRANSFORMATION    OF   COORDINATES. 


40.  When  the  eruation  of  a  curve  referred  to  one  system  of 
coordinates  is  known,  it  is  often  desirable  to  obtain  the  equation 
of  the  same  curve  referred  to  a  different  system.  We  shall  be 
able  to  find  formulas  which  connect  the  coordinates  of  any  point 
in  the  plane  in  Xhejirst  system  with  those  of  the  same  point  in 
the  s£co?id,  and  thus  render  easy  the  desired  change. 

41.  We  will  first  compare  the  coordinates  of  a  point  referred 
to  one  set  of  axes  with  the  coordinates  of  the  same  point  referred 
to  a  parallel  set. 

Let6>Xand  OY 
be  the  original  set 
of  axes  ;  O'  X'  and 
O'  y  the  new  set, 
parallel  respective- 
ly to  the  old. 

O',  the  new  ori- 
gin, has  coordi- 
nates in  the  old 
system  which  we 
will  call  x^  and  y^. 

Then, 


Y 

Y 

P 

0 

M 

X 

0 

M,                              1 

VI 

X 

Let  P  represent    any   point 
referred  to  XO  Fare 


in    the    plane  ;    its   coordinates 


::} (3) 


62  ANALYTIC    GEOMETRY. 

X  =  OM,    y  r=.MF, 

Referred  to  X^  O'  V,  the  coordinates  of  I'  are  O'  M'  and 
M^  P;  these  we  will  call  x^  and  y^  respectively,  to  distinguish 
them  from  x  and_>^;  but  we  must  remember  that  x^  and  y  are 
here  variables  in  the  sa?ne  sense  that  x  and  y  are  variables. 

From  the  figure  and  §  2  we  have 

C>J/=  OM^  +  M^M=.  OM,  +  O^ M^    •  •  •  •  (i) 
MP  =  MM'  +  M'P=  3f,0'  +  M'P.    ....  (2) 

Substituting  in  these  equations  the  values  of  the  lines  as  given 
above,  we  have 

X   =  Xq   -{-  X' 

y  =  yo  +  y' 

which  are  evidently  the  equations  sought,  since  they  enable  us  to 
find  values  for  either  pair  of  coordinates  for  P,  in  terms  of  the 
other  pair  and  the  constants  x,  and  y,  which  determine  the 
relative  positions  of  the  two  sets  of  axes. 

We  can  best  illustrate  the  use  of  these  formulas  by  an 
example. 

The  equation  of  a  line  referred  to  rectangular  axes  is 

2x  +  sy^5', 
find  the  equation  of  the  same  line,  referred  to  a  parallel  system 
which  has  the  point  (2,  4)  for  its  origin. 
Here 

Xq  =  2  and  y,  =  4. 

The  equations  for  transformation  are,  therefore, 

X  =  2  +  x',    y  =  4  -\-  y. 

These  equations  mean  that  a/iy  point  in  the  plane  has  its 
abscissa  in  the  Jirsf  system  greater  than  its  abscissa  in  the  second 
by  two  units  of  length,  and  that  the  old  ordinate  exceeds  the  new 
hy  four  units. 


TRANSFORMATION    OF    COORDINATES.  63 

Now  the  locus  of  the  given  equation  comprises  all  the  points 
which  have  twice  their  old  abscissas  plus  thrice  their  old  ordinatcs 
equal  to  five  ;  consequently  it  consists  of  those  points  which 
have  twice  the  new  abscissa  increased  by  two  plus  thrice  the  new 
ordinate  increased  by  four  equal  to  five. 

Its  equation  in  the  new  system  is,  therefore, 

2  (^'  +  2)  +  3  {y^  +  4)  =  5, 
or 

2;v'  +  4  +  3/  +  12  -  5, 
or 

2  ^'  +  3  y  +  II  =  o. 

Since  in  this  equation  the  accents  are  only  used  to  distinguish 
the  new  coordinates  from  the  old,  now  that  we  have  changed  to 
the  new  system  and  there  is  no  longer  opportunity  for  confusion, 
we  will  use  the  usual  symbols  for  the  coordinates  of  the  point 
which  generates  the  locus,  and  write  the  equation 

2^  +  37+  11=0. 

The  student  should  construct  the  two  sets  of  axes,  and  then 
plot  the  loci  of  the  two  equations,  and  show  that  these  loci  are 
one  and  the  same  straight  line. 

Note.  —  Since  the  results  of  this  section  have  been  obtained 
by  the  principle  of  §  2,  and  without  reference  to  the  angle  X  O  V, 
the  same  formulas  may  be  used  for  transforming  from  one 
oblique  system  to  2,  parallel  oblique  system,     [v.  §  8.] 

42.  We  will  next  find  the  formulas  by  which  we  may  change 
from  one  rectangular  system  to  another,  keeping  the  same  oj-igin. 

Let  OX  and  6>  F  be  the  original  axes;  O X^  and  O  V\  the 
new  axes.  Let  the  angle  XO  X,  which  the  new  axis  of  x  makes 
with  the  old,  be  called  i>. 

The  coordinates  of  P  (any  point  in  the  plane)  in  the  old 
system  are  x  -  O  M  and  y  -  MP;  in  the  new  system  they 
are  x^  ^  O  M'  and  /  =  M'  P.     From  M'  draw  M'  P  perpen- 


64 


ANALYTIC   GEOMETRY. 


dicular  to  O  X,  and 
J/'  T  parallel  to  O  X. 
Let  the  latter  line  (or 
its  extension)  meet 
MP  (or  its  extension) 
in  S.  Take  U  on  the 
line  M^  P  so  that  the 
angle  X^  M'  U  shall 
be  90°  of  positive  ro- 
tation. T  and  X  are, 
of  course,  to  be  chosen 
sothatJ/Tandil/'^' 
shall  have  the  direc- 
tions of  O  X2.x\6.  OXK 

From  the  figure  and  §  3,  we  have 

TM'U=  TMX>  +  X'M'V 

=  {f  +  90°. 

The  triangles  O  R  M'  and  M'  SP  are  therefore  triangles  of 
reference  for  if  and  (90°  +  i>),  respectively. 
Now,  from  the  figure  and  §  2,  we  have 

x=OM=OR  +  RM^OR+M'S....{i) 
y  =  AfP=  AfS+  S  P  =RM'  +  SP (2) 


But 

O  /?   =  O  Af  cos  0  =  x'  cos  {y 

M'  S  =  M'  Pcos  (90°  +  d)  =  -  M'  Ps\n  if  =  -/  sin  & 

RAP  =  OAf'  sin  0^  =  x'  sin  (f 

.S^  P   =  Af'P  sin  (90°  +  {y)  =  M'  Pcos  r^  =  /  cos  &    . 

Substituting  in  equations    (i)   and    (2)   the  values  found  in 

(3)-(6),  we  have 

X  =  x'  cos  i>  —  y  sin  & 
y  =  a;'  sin  i*)-  +  y'  cos 

which  are  the  required  equations  of  transformation,  since  they 


'.[ 


(3) 
(4) 
(5) 

(6) 


(7) 


TRANSFORMATION    OF    COORDINATES.  65 

connect  the  old  and  new  coordinates  by  means  of  the  arbitrary 
constant  &  which  determines  the  relative  positions  of  the  two 
sets  of  axes. 

EXAMPLE. 
The  equation  of  a  curve  is 

x'+f  =  a'; '.  (I) 

find  the  equation  of  the  same  curve  when  referred  to  axes  which 
make  an  angle  of  30°  with  the  original  axes,  but  retain  the  same 
origin. 
Here 

^  =  30° ;   sin  ^  =  -;   cos  ^  =  ^. 
2  2 

The  formulas  for  transformation  are,  therefore, 

\/'X  I  I  V  "? 

X  =  x'  —  -  y     and    y  z=  -  x'  +  y'. 

2  2  2  2 

Since  the  coordinates  of  any  point  of  the  curve,  when  referred 
to  the  old  system,  must  satisfy  equation  (i),  their  equivalents  in 
terms  of  the  new  coordinates  of  those  points  must  satisfy  equa- 
tion (i)  ;  therefore  we  have 


or 


3^'2_  2\/^x'y'  +y^  ,  x'"^  -f  2  \^3x'y  -f  3/^  _    2 


4 
which  reduces  to  the  form 

x'^  +y'  =  a' (3) 

If  we  drop  the  accents  —  because  we  have  now  left  the  old 
system  and  need  no  longer  fear  a  confusion  of  the  old  and  new 
coordinates  of  any  point  —  we  may  write  equation  (3) 

x"^  +  y^  =  a"" (4) 


66 


ANALYTIC   GEOMETRY. 


Since  (4)  is  identical  in  form  with  (1),  we  see  that  the  curve 
has  the  same  law  governing  the  coordinates  of  its  points  when 
referred  to  the  new  axes  as  when  referred  to  the  old.  This  is 
evidently  true  from  the  geometrical  properties  of  the  curve ;  for 
the  locus  of  (i)  is  a  circle  with  its  centre  at  the  origin,     [v.  §  16.] 

Corollary.  —  The  formulas  of  §42  become  much  simplified 
when  {^  =  90°.      In   this  case  sin   i>  =  i   and  cos  {>  =  o  :    the 
equations  for  transformation  therefore  become 
X  —  —  y     and    y  =  x'. 

We  may  obtain  the  same  result  by  considering  that  in  this  case 
we  merely  interchange  the  axes  of  x  and_>^/  at  the  same  time 
changing  the  positive  direction  of  the  horizontal  axis. 

43.  To  change  from  a  rectangular  to  a  polar  system,  with  the 
origin  for  pole^  and  the  axis  of  x  for  initial  line. 

Let  P  be  any  point  in 
the  plane.  Its  coordinates 
in  the  rectangular  system 
are 

X  ^  OAf,y  =  MP; 
in  the  polar  system,  they 
are 
x"       r=OP,cj>  =  XOP; 
O  M  P  is   a   triangle  of 
reference  for  <J3 ;  therefore  we  have 

O  M  =  O  P  X  cos  <^,  or  X  =  r  cos  cj>y (i) 

M P  =  O  P  X  sm  (f),  or  y  =  r  sm  ff>^      .   .  .   .   (2) 

which  are  the  required  equations  for  transformation,  because 
they  connect  the  rectangular  and  polar  coordinates  of  any  point 
in  the  plane. 


TRANSFORMATION   OF   COORDINATES.  67 

EXAMPLE. 
Find  the  polar  equation  of  the  circle 

referred  to  the  centre  as  pole. 

We  have  seen  in  the  example  of  §  42  that  the  position  of  the 
axis  of  X  does  not  change  the  equation  of  this  curve,  provided 
the  centre  is  origin.  Therefore,  if  the  centre  is  the  pole,  what- 
ever the  direction  of  the  initial  line,  the  formulas  for  transforma- 
tion are 

X  =  r  cos  (f),  y  =  r  sin  <^. 

Substituting  for  x  and  y,  in  the  equation  of  the  circle,  their 
values  in  terms  r  and  <^,  we  have 

r^  cos^  (fi  +  r^  sin^  </>  =  a^, 
or 

r^  (cos^  </)  +  sin^  </>)  =  a% 
or 

which  may  be  written 

r  =  ±  a, 

a  Jesuit  which  agrees  with  the  polar  equation  of  the  circle  found 

in  §  38. 

The  double  sign  is  explained  by  the  fact  that,  with  any  value 
of  </),  the  radius  vector,  measured  in  either  the  positive  or  the 
negative  direction,  meets  the  curve  at  a  distance  a  from  the 
pole. 

44.  In  §§  41-43,  we  have  obtained  the  equations  for  trans- 
forming coordinates  in  three  simple  cases.  By  combining  these 
formulas,  however,  we  can  change  from  a  rectangular  system  to 
any  other  rectangular  or  polar  system,  or  luce  versa  ;  for  we  can 
first  change  the  origin  to  any  desired  point,  by  §  41  ;  then,  by 
§  42,  make  the  axes  take  any  direction  ;  and  lastly,  if  desired, 
change  to  polar  coordinates,  by  §  43. 


68  ANALYTIC    GEOMETRY. 

There  is  an  important  general  principle  concerning  the 
transformation  of  coordinates  which  we  will  now  establish. 

Suppose  the  equation  of  a  certain  curve  is  an  algebraic 
equation  of  the  nih  degree  in  the  rectangular  coordinates 
x,j:  and  suppose  we  transform  to  a  new  system  of  rectangular 
coordinates  x\  y'  situated  in  any  manner  whatever.  The 
equation  of  the  curve  may,  and  usually  will,  take  on  a  form 
very  different  from  what  it  had  at  ^x's,\.,h\x\.  it  will  still  be  an 
algebraic  equation  of  the  nth  degree  in  the  coordinates  x\  y' . 
In  order  to  prove  that  this  statement  is  correct,  let  us  consider 
how  it  is  we  pass  from  the  original  form  of  the  equation  of  the 
curve  to  the  new  form.  For  this  purpose  we  must  first  apply 
formulce  (3),  §41,  in  order  to  bring  the  origin  to  the  desired 
position  and  then  formulae  (7),  §  42.  In  each  of  these 
formulae  the  old  coordinates  are  expressed  as  polynomials  of 
the  first  degree  in  terms  of  the  new.  These  transformations 
will,  therefore,  leave  the  equation  in  the  form  of  an  algebraic 
equation  of  degree  no  higher  than  n.  It  is  not  entirely  obvi- 
ous, however,  that  the  degree  of  the  new  equation  might  not 
in  some  cases  be  less  than  n  through  the  cancellation  of  the 
terms  of  highest  degree.  We  have,  therefore,  as  yet  merely 
shown  that  the  degree  of  an  algebraic  equation  cannot  be 
raised  by  a  change  from  one  system  of  rectangular  coordinates 
to  another.  Suppose,  now,  that  the  degree  of  the  equation 
could  be  lowered.  Then  starting  with  the  equation  in  x' ,  y\ 
let  us  transform  back  to  the  original  system  of  coordinates 
X,  y.  This  transformation  must  evidently  bring  us  back  pre- 
cisely to  the  equation  we  started  with,  so  that  if  the  degree  of 
the  equation  had  been  lowered  when  we  passed  from  the  first 
system  of  coordinates  to  the  second,  it  would  be  raised  when 
we  pass  back  from  the  second  system  to  the  first  ;  and  this  is 
impossible,  since,  as  we  just  saw,  no  transformation  of  coordi- 
nates can  raise  the  degree  of  an  equation. 


TRANSFORMATION  OF  COORDINATES.  69 

We  see,  therefore,  that  the  degree  of  the  equation  of  a  curve 
depends  on  the  curve  itself,  and  not  at  all  on  the  special  sys- 
tem of  rectangular  coordinates  used.  This  fact  justifies  us  in 
classifying  curves  according  to  the  degree  of  their  equations, 
and,  in  fact,  it  is  customary  to  speak  of  curves  of  the  first 
degree,  of  the  second  degree,  etc.  We  have  already  seen  that 
the  only  curve  of  the  first  degree  is  the  straight  line.  It  is  with 
the  study  of  curves  of  the  second  degree  that  we  shall  be 
concerned  from  now  on. 

Besides  algebraic  curves,  which  we  classify  according  to 
their  degree,  there  are  also  transcendental  curves  whose  equa- 
tion in  rectangular  coordinates  cannot  be  written  in  algebraic 
form.  Examples  of  such  curves  will  be  found  in  Ex.  (9)- 
(12),  p.  19. 

The  following  examples  may  be  solved  by  the  application  of 
one  or  more  of  the  methods  of  transformation  explained  in 
§§  41-43- 

EXAMPLES, 
(i.)    The  equation  of  a  curve  is 

;c^  +  y  __  4^  _  6  7  =  18  j 

find  the  equation  of  the  same  curve,  referred  to  axes,  parallel  to 
the  old,  through  the  point  (2,  3). 

Ans.  AT^  +  _>'^  =  31. 
(2.)    The  equation  of  a  curve  is 

what  is  its  equation   when   referred   to   lines  which  bisect  the 
angles  between  the  old  axes  ? 

Ans.  X  y  =  2)' 
(3.)    When  we  change  from  one  rectangular  system  to  another 
without  changing  the   origin,  x^  +  f-  must  equal  ^^  +y^  be- 


70  ANALYTIC    GEOMETRY. 

cause  each  denotes  the  square  of  the  distance  of  the  same  point 
from  the  origin.     Verify  this  by  squaring  and  adding  the  expres- 
sions for  X  and_y,  in  terms  of  x'  and  j',  given  in  §  42. 
(4.)    The  equation  of  a  curve  is 

^xy  -  ^  x'^  z=i  a^ ; 

turn  the  axes  through  an  angle  whose  tangent  is  2,  and  find  the 
new  equation  of  the  curve. 

Ans.  xP"  —  4^^  =  a^. 

(5.)    Find  the  polar  equation  of  the  line 

Ax-\-By-\-C-=o^ 

when  the  origin  is  the  pole  of  the  new  system,  and  the  axis  of  x 
the  initial  line. 

Ans,  {A  cos  (/)  +  -^  sin  <^)  r  4-  C  =  o. 

(6.)    Under  similar  circumstances,  find  the  polar  equation  of 
the  curve  which  has  for  its  rectangular  equation 

X"  V2 

Ans.  r^  = 


a^  sin^  ffi  -\-  d^  cos^  <^* 
(7.)    The  equation  of  a  curve  is 

2  ^' +  3  / -16^+18^  + 53  =  0; 

find  its  polar  equation,  when  (4,  —  3)  is  the  pole,  and  the  initial 
line  is  parallel  to  the  axis  of  x. 

Ans.  r^  -  — r^. ^— ■ . 

3  sin''  9  +  2  cos''  </) 

(8.)  By  combining  the  results  of  §§  41,  42,  obtain  formulas 
for  changing,  at  once,  from  one  rectangular  system  to  any  other 
rectangular  system. 


CHAPTER  V. 


THE   CIRCLE. 


45.  A  circle  may  be  defined  as  the  path  of  a  point  which 
moves  in  a  plane  so  that  its  distance  from  a  fixed  point  in  that 
plane  is  constant. 

Let  us  find  the  equation  of  the  circle  ;  i.  e.  an  equation  whose 
locus  must  be  a  circle,  but  which  may  in  turn  represent  all 
circles. 

In  the  figure,  C  is  the 
fixed  point,  with  coordi- 
nates O  J/c  =  ^c  ^"d 
J/c  C  —  }'&'■>  Pf  with  coor- 
dinates O  M  —  X  and 
M  P  ^  y,  is  the  point 
which  generates  the 
curve. 

Letting     a     represent 
X    the  constant  distance  of 
P  from    C,  we  may  ex- 
press the  definition  of  the  circle  by  the  equation 

CP=a , 


(i) 

By  §  10,  the  distance  between  the  two  points  C  and  P  may  be 
expressed  in  terms  of  their  coordinates,  and  we  have 


72  ANALYTIC    GEOMETRY. 

Substituting  this  value  in  (i),  we  have 


and  clearing  of  radicals, 
,  (x  -  x,y  -{.  (j  -  je)^  =  ^^ (2) 

which  must  be  the  equation  of  the  circle,  for  it  expresses  in  an 
equation,  connecting  x  andj^,  the  essential  property  of  the  circle 
generated  by  the  point  {x,  y). 

Corollary  1 .     When  the  centre  is  at  the  origin 
Xc  =  o  and  y^  =  ^  i 
the  equation  becomes  in  this  case 

x^  +y  =  a^, 
a  result  already  obtained  in  §  16. 

Corollary  2.     When  a  diameter  is  taken  as  the  axis  of  x^ 
and  its  left-hand  extremity  as  the  origin, 

Xc  =  a  and  y^  =  o ; 

in  this  case  the  equation  of  the  circle  is 

{x  -  ay  +  /  =  a% 


or. 


x^  -\-  y^  =  2  a  X. 


EXAMPLES. 

(i.)  Write  the  equation  of  the  circle  whose  radius  is  5,  and 
which  has  for  its  centre  the  point  (3,  4)  ;  reduce  the  equation, 
and  show  from  its  form  that  the  circle  passes  through  the  origin. 
(2.)  A  circle,  with  radius  4,  has  its  centre  on  O  X  and  is 
tangent  to  O  Y :  what  is  its  equation  if  it  lies  on  the  right  of 
O  Y?     What  if  on  the  left? 

y^  -^x  r^  o, 
y'^  -{-  d>  X  =  o. 
(3.)    Obtain  the  equation  of  Cor.  i,  from  the  first  form  of  the 


Ans.  ^     o        2 


THE    CIRCLE.  ^  73 

equation  of  the  circle,  by  transforming  the  axes  to  a  parallel 
set  through  the  point  (^o^c)* 

(4.)  Obtain  the  equation  of  Cor.  2  from  each  of  the  preced- 
ing forms,  by  transformation  of  coordinates. 

(5.)  Prove  analytically  that  any  angle  inscribed  in  a  semi- 
circle is  a  right  angle. 

Suggestion. — Take  the  diameter  bounding  the  semicircle 
as  axis  of  x. 

(6.)  Find  the  equations  of  the  three  circles  whose  diameters 
are  the  sides  of  the  triangle  formed  by  the  lines 

2x  -\-  y    -7=0, 
X-  y    +1=0. 


x' 

+/ 

-  5  ^ 

-  4  i'  +  9 

= 

0, 

x^ 

+/ 

-3-^ 

-5J  +  8 

— 

0, 

x" 

+/ 

-  ^x 

-  3  J  +  5 

= 

0. 

(7.)    Find  the  equation  of  the  circle  whose  centre  is  the  ori- 
gin, and  which  is  tangent  to  the  line 

2-^-7  +  3=0' 

Ans.  5  (^2^/)  =  9. 

(8.)    The  equation  of  a  chord  of  the  circle 

x'^  -\-  y'^  =  100 
is 

1  X  -\-y  =  50; 

find   the   equation   of   the   circle  which   has  this  chord  for  a 
diameter. 

(9.)    Find  the  points  of  meeting  of  the  circles 
x'^  +  y-  —   2  X  —  /^  y  —   20   =0, 

:v^  +y  —  14  a"  -  i6_y  +  100  =  o. 

Ans.  (4,  6),  (5,  5). 


74  ANALYTIC    GEOMETRY. 

(lo.)    Find  the  equation  of  the  chord  common  to  the  circles 
given  in  the  last  example,  and  show  that  this  chord  is  perpen- 
dicular to  the  line  joining  their  centres. 
(ii.)    Show  that  the  circles 

(^  -  2)'  +  (y-  3)'  =  36. 
(x  -  -,2)'  +  (y  -  ^y  =  i6, 

are  tangent  at  the  point  (8,  3). 

(i  2.)  Prove  that  the  perpendicular  upon  a  diameter  of  a  circle 
from  any  point  of  the  circumference  is  a  mean  proportional  be- 
tween the  segments  of  the  diameter. 

(13.)  Prove  that  if  two  circumferences  are  tangent  internally, 
and  the  radius  of  the  larger  is  the  diameter  of  the  smaller,  any 
chord  of  the  larger  drawn  from  the  point  of  contact  is  bisected 
by  the  circumference  of  the  smaller. 

Suggestion.  —  Take  the  point  of  contact  for  the  origin,  and 
a  diameter  for  the  axis  of  x ;  the  equation  of  any  chord  may 
then  be  written 

y  =  \  X. 

(14.)  If  two  straight  lines  are  drawn  through  the  point  of 
contact  of  two  circles,  the  chords  of  the  intercepted  arcs  are 
parallel.  Prove  when  the  circles  are  tangent  externally,  taking 
the  origin  at  the  point  of  contact. 

46,  Equation  (2)  when  expanded  becomes 

This  equation  has  the  form 

x'  +/  +  ax  +  by  +  c=o (i) 

where  a,  b,  c  are  constants.     The  circle  is  therefore  a  curve  of 
the  second  degree. 

Conversely  any  equation  of  the  form  (i)  can  be  shown  to 
represent  a  circle,  provided  its  locus  is  a  curve  of  any  sort.* 

*  The  reason  for  this  restriction  will  be  apparent  presently. 


THE    CIRCLE.  75 

Before  proving  this  fact  we  will  illustrate  it  by  a  numerical 
example.     Consider  the  equation 

^2  ^  y  _  ^j^  _|_  ^  y  —  I  =o. 

This  equation  is  of  the  form  (i),  and  we  will  prove  that  it  repre- 
sents a  circle.,  and  at  the  same  time  find  the  centre  and  radius  of 
this  circle  by  the  following  device.  We  arrange  the  terms  of 
the  equation  in  three  groups.  First,  the  two  terms  involving 
.T,  then  two  terms  involving  y,  and  then,  transposed  to  the 
right-hand  side,  the  constant  term;  thus: 

x^  —  4x         +  y  +  4>'         =  !• 

Now  apply  to  the  first  group  of  terms  the  process,  familiar 
from  algebra,  of  completing  the  square  by  adding  the  square  of 
half  the  coefficient  of  x.  In  the  same  way  complete  the 
square  for  the  second  group  of  terms  by  adding  the  square  of 
half  the  coefficient  of ;'.  The  two  quantities  which  we  have 
just  added  to  the  left-hand  side  of  the  equation  must,  of 
course,  also  be  added  on  the  right,  and  we  thus  get 

(x  -  2y  -{-  (y  +  2y  =  9. 

This,  however,  is  [§  45,  (2)]  the  equation  of  the  circle  with 
centre  at  (2,  —  2)  and  radius  3. 

Precisely  this  method  of  completing  the  square,  which  we 
have  here  applied  in  a  numerical  case,  can  be  applied  without 
change  to  the  general  equation  (i),  and  gives 

Provided   the  right-hand  side  of  the  equation   is  positive,  we 

/a          ^\ 
have  here  the  equation  of  the  circle  with  centre  at  ( , ) 

and  radius  \l—  c  -\ -\ .        If,    however,    the    right-hand 

\  4        4 

side  of  (2)  is   negative,  this  reasoning    is  no  longer  possible, 


76  ANALYTIC    GEOMETRY. 

since  the  radius  we  should  thus  obtain  is  an  imaginary  quan- 
tity. In  fact  in  this  case  equation  (2)  has  no  locus,  i.e.  there 
is  no  real  point  whose  coordinates  satisfy  this  equation.  For 
the  left-hand  side  of  (2),  being  the  sum  of  two  squares,  is  essen- 
tially positive,  while  the  right-hand  side  is  now  supposed 
negative. 

There  remains  to  be  considered  the  case  in  which  the  right- 
hand  side  of  (2)  is  zero.     In   this  case  the  circle  consists  of 

a  single  point  ( ,  —  —  j,  since  the  left-hand  side  of  (2), 

being  the  sum  of  two  squares,  can  vanish  only  when  each  of 
these  squares  vanishes.  This  point  is  commonly  regarded  as 
a  circle  of  radius  zero. 

Summing  up,  we  may  say  that  equation  (i)  represents  a 
circle,  a  point,  or  has  no  real  locus  according  as  the  right- 
hand  side  of  (2)  is  positive,  zero,  or  negative.  It  is  usual  to 
designate  these  three  cases  by  saying  we  have  respectively  a 
real,  null,  or  imaginary  circle. 

The  general  equation  of  the  second  degree  in  x  and_)'  is 
much  more  general  than  equation  (i),  for  not  only  may  the 
terms  in  x^  and^  have  coefficients  different  from  unity,  but 
there  may  be  a  term  involving  the  product  xy.  Thus  the 
general  equation  of  the  second  degree  may  be  written 

Ax"  +  Bxy  +  Cf  +  £>x  +  Ey  +  F  =  o.  .  ...   (3) 

This  equation  usually  represents  a  curve  which  is  much  more 
complicated  than  a  circle.  It  will  represent  a  circle  (real, 
null,  or  imaginary)  ii  B  =  o  and  ^  =  C  ^^  o,  for  then  (3)  has 
the  form 

Ax""  +  Ay""  +  Dx  +  Ey  +  F  =  o, 

and   this  equation  reduces  to  (i)  if  we  divide  it  through  by  A 


THE    CIRCLE.  77 

EXAMPLES. 
Find  the  centres  and  radii  of  the  following  circles  : 
(i)  A""  +  jj''  +  6  ^  —  4  jv  4-  9  =  o.  Ans.  (—  3,  2),  2. 

(2)  X''  -f  j'  —  6^  —  10;'  +  34  =  o.  Ans.  (3,  5),  o. 

(3)  ^'  +/-2,x+y  +  2  =  o.  Ans.  (|,  -  -^)  ,  |  V~2' 


I       2      1 
(4)    \2  x"^  ■\- 12 y^ -\- \(i  X  —  \2 y -\- 7^  ^^  o.    Ans.   1 ,  — 


2      1  \       2 
3" 


47.  We  know  from  Geometry  that  a  circle  is  determined  by 
any  three  points  on  the  circumference.  We  may,  therefore, 
obtain  the  equation  of  a  circle  which  passes  through  any  three 
given  poifits. 

Let  ^p  P^,  and  P^  be  the  three  given  points. 

Since  equation  (2),  §  45,  represents  in  turn  all  circles,  it  will 
represent  the  required  circle  when  we  give  appropriate  values  to 
the  arbitrary  constants,  x^,  y^,  and  a,  in  that  equation. 

That  the  locus  of 

may  pass  through  P^^  P^  and  Py,  it  is  necessary  and  sufficient 
that  the  coordinates  of  these  points  should  satisfy  equation  (i)j 
these  conditions  are  expressed  by  the  three  equations  :  — 

(pc^-x^y  ^  {y^  -y^y  =  a% 

{x^^x-.y-h  {y,  -y^y  =  a\ 

In  these  equations  the  arbitrary  constants,  x^,  jc,  and  a,  are 
the  quantities  whose  values  we  seek,  while  the  other  quantities 
are  the  known  coordinates  of  the  given  points.     We  have,  then, 


78  ANALYTIC    GEOMETRY. 

three  equations  between  three  unknown  quantities,  and  can  deter- 
mine them  all  by  elimination. 

Substituting  in  (i)  the  values  thus  found  for  x^,  yc)  ^nd  a,  we 
have  the  equation  of  the  required  circle. 

EXAMPLES. 

(i.)  Find  the  equation  of  the  circle  which  passes  through  the 
points  (9,  6),  (10,  5),  (3,  -  2). 

Calling  these  points  J^^,  P.^,  and  Z^^,  the  three  equations  from 
which  we  may  obtain  the  proper  values  of  x^,  j'c,  and  a  are 

(9-^„)^+     (6_j,)=  =  «^ (i) 

(io-x,)^+      (5_jg^=«^ (2) 

(3-^c)'+(-2-a)^  =  «' (3) 

We  can  easily  eliminate  a  from  these  equations,  and  obtain 
two  new  equations  connecting  x^  and  y^.^  from  which  we  shall 
find 

x^  =  6,  7c  =  2. 

Substituting  these  values  in  either  of  the  equations  (i),  (2), 
(3),  we  obtain 

a"  =  25. 

We  have  now  determined  all  the  constants  in  the  required 
equation,  which  must  be 

(^-6)«+(^-2)«=2S, 

or 

0^  ^y^  -  12  :v  -  4>'  +  15  =  o. 

(2.)   Find  the  circle  through  (2,  -  3),  (3,  -  4),  (  -  2,  -  i). 

Ans,  x^  +y^  +  S  X  +  207  +  31  =  0. 

(3.)  Find  the  equation  of  the  circle  circumscribed  about  the 
triangle  given  in  §  20,  Ex.  7. 

Ans,  ^x^  +  ^y  —  13  jp  —  II 7  +  20  s=  o. 


THE    CIRCLE.  79 

(4.)  Solve  the  last  three  problems  by  starting  from  equation 
(i)  of  §  46. 

48.  The  term  secant  is  used  in  Analytic  Geometry  to  denote 
a  straight  line  which  cuts  a  curve  in  two  or  more  points. 

If  one  of  the  points  in  which  a  secant  cuts  a  curve  is  moved 
along  the  curve  towards  a  second  of  these  points,  the  secant 
approaches  the  position  of  a  tangent  to  the  curve  at  the  second 
point,  and  it  is  by  thus  regarding  the  tangent  as  the  limit  of  a 
secant  that  we  are  able  in  Analytic  Geometry  to  obtain  its 
equation. 

A  normal  to  a  curve  at  any  point  is  the  straight  line  perpen- 
dicular to  the  tangent  to  the  curve  at  that  point. 

As  it  is  often  necessary  to  consider  tangents  and  normals  to 
curves,  we  shall  seek  the  equations  of  these  lines  for  each  curve 
that  we  study. 

49.  Let  us  find  general  forms  for  the  equations  of  a  tangent 
and  a  normal  to  a  given  circle  at  a  given  point  of  the  curve. 

We  will  use  the  simplest  form  of  the  equation  of  the  circle  :  — 

^^ +/  =  <'" (i) 

and  let  P^  be  the  given  point. 

By  §  26,  the  equation  of  any  line  through  P^  may  be  written 

y  -yx^^  (^-Xi) (2) 

Equation  (2)  will  represent  a  secant^  a  tangent,  or  a  normac\ 
according  to  the  value  given  to  A. 

Let  Ag  represent  the  slope  of  a  secant  which  cuts  the  curve 
in  P^  and  a  second  point  P^\  A^,  the  slope  of  the  tangent  at 
P^\  Ajj,  the  slope  of  the  normal  at  Py 

If  we  obtain  values  for  Ag,  A^,  and  A^.,  in  terms  of  known 
quantities,  and  substitute  them,  in  turn,  in  equation  (2),  we 


8q 


ANALYTIC    GEOMETRY. 


shall  have  the  equations  of  the  secant,  tangent^  and  normal^ 
respectively. 

Instead  of  denoting  tlie  coordinates  of  P,_  by  {_x^^y^  it  will  be 
convenient  to  introduce  the  following  notation: 

Let  M ^  and  M^  be  the   feet  of    the  ordinates  of  P^  and  P^ 


V 

] 

/     1    ^\ 

^2 

0 

respectively,  and  draw  through  P^  a  line  parallel  to  the  axis  of 
A'  cutting  J/,  P^  in  N,  and  let 


P^N=h,         NP^  =  k, 


According  to  this  notation  the  coordinates  of  P^  may  be  writ- 
ten (jCj  -j-  //,  y\  -\-  k).  The  student  should  satisfy  himself  by 
drawing  other  figures  that  this  is  true  for  all  possible  positions 
of  P^  and  P^.     The  slope  of  the  secant  is 


K  = 


k 


(3) 


Now  the  difficulty  of  our  problem  consists  in    this,  that  as 
jP,  moves  down    the   circle   towards  P^    both   numerator  and 


THE    CIRCLE. 


8i 


denominator  of  this  fraction  approach  zero  as  a  limit,  and  it  is 
therefore  not  possible  to  see,  without  further  examination, 
what  limit  the  fraction  itself  is  api)roaching.  This  is  not  sur- 
prising, as  we  have  not  yet  made  any  use  of  the  fact  that  F ^ 
and  P^  are  both  on  the  circle  (i).  This  fact  gives  us  the  two 
equations  of  condition: 


{x,  +  hf  +  (j-    +  Iz)^  =  a- 


{.) 


By  subtracting  the  first  of  these  equations  from  the  second  we 
get  the  relation 

2  /^  ^1  +  /^^  -f  2  ky^  +  k'  =  o. 

k 
In  order  to  get  another  expression  for  the  fraction  —    we 

have  merely  to  divide  this  equation  by  /i  : 

2  Jt',    +   /^  +    2 


or 


2  x^  +  /i  +  (2  ;',  +  ^) 


Accordingly, 


We  thus  have 


A.=  - 


2  X, 


(5) 


2  ;',  +  ^ 

UNIVERSfTY  OF  CALIFORNIA 

DEPARTMENT  OF  CIVIL  ENGINEERING 

BERKELEY,  CALIFORNIA 


82  ANALYTIC    GEOMETRY. 

This  expression  for  Ag  is  less  simple  than  that  given  by  equa- 
tion (3),  but  it  has  the  advantage  of  being  in  a  form  in 
which  we  can  easily  determine  its  limit.  As  P^  approaclies 
7^,,  both  h  and  k  approacli  zero,  so  that  the  numerator  of  the 
fraction    in   (5)  approaches  2  :r,,  the  denominator   2  }\.     The 

2  X 
slope  Ag  itself  therefore  approaclies  the  limit  —    — ^  .      There- 
fore 

^.=  -5 (6) 

Since  the  normal  is  perpendicular  to  the  tangent,  vi^e  have,  by 
§  2>T,,  Cor.  2, 

A.  =  -f  =  f (7) 

At  X^ 

The  equations  of  the  required  tangent  and  normal  are,  there^ 
fore, 

.>'-.>'i=  -y|(^-^i), (8) 

and 

y  -yx  =  JC-^  -^1) (9) 

Clearing  (8)  of  fractions,  we  have 


y^y  - y^  ^  - x^x  ^r  x'' 


1    y 


transposing  and  reducing  by  means  of  equation  (4),  we  have 

x^x  +y^y  =  a^, (10) 

the  simplest  form  of  the  equation  of  the  tangent. 
Reducing  equation  (9),  we  have 

y^x  -  x,y  =  o, (11) 

the  equation  of  the  normal. 

The  form  of  equation   (11)   shows  us  that  the   normal   must 


THE    CIRCLE.  83 

pass  through  the  origin  (v.  §  19).  Now  the  origin  is,  in  this 
case,  the  centre  of  the  circle,  so  that  we  have  proved  the  well- 
known  theorem  of  geometry,  that  a  perpendicular  to  a  tangent 
to  a  circle,  at  the  point  of  tangency,  passes  through  the  centre. 

Note.  This  method  of  finding  the  equations  of  tangent  and 
normal  should  be  carefully  studied,  not  only  on  account  of  the 
importance  of  the  tangent  and  normal  to  the  circle,  but  because 
it  will  be  used  in  finding  the  equations  of  these  lines  for  each  of 
the  curves  which  we  shall  study.  Equation  (i)  will  change  as 
we  change  our  curve,  and  so  equations  (4)  and  (5)  will  have 
different  forms,  and  the  algebraic  work  by  which  we  obtain  the 
value  of  Ag  will  vary;  but,  in  the  essential  steps,  the  same  method 
may  be  used  to  find  the  equations  of  the  tangent  and  normal  to 
any  curve. 

EXAMPLES. 

(i.)  Find  the  equations  of  the  tangent  and  normal  to  the 
circle 

x^  ■\-  f"  ^  25 
at  the  point  (3,  —  4). 

The  given  point  is  a  point  of  the  circle,  because  its  coordi- 
nates satisfy  the  equation  ;  these  coordinates  therefore  corre- 
spond to  .Tj  and  j/j  in  the  general  equations  of  tangent  and 
normal,  while  d^  —  25. 

The  equations  sought  are,  therefore, 

3^  -  4j'  =  25, 
and 

_4^_3j  =  o, 
respectively. 

(2.)    Find  the  equations  of  tangents  and  normals  to  the  circle 
x^  +  yr-  —  1 69 
at  all  points  of  the  curve  which  have  their  abscissas  numerically 
equal  to  five. 

(3.)    Find  the  equation  of  the  normal  to  the  circle,  at  a  point 


84  ANALYTIC    GEOMETRY. 

jP,,  by  means  of  the  fact  that  it  passes  through  the  centre ;  find 

the  equation  of  the  tangent  from  that  of  the  normal  by  means  of 
the  mutual  relation  existing  between  these  lines. 

(4.)    Find  the  equations  of   the  tangent  and   normal   to  the 

circle 

x^  -\-  y  =  2  a  X 

at  a  point  J^^  of  the  curve. 

ij\x-  (x^  -a)y-ay,  =  o. 

(5.)  Confirm  these  results  from  equations  (10)  and  (11),  by 
transformation  of  coordinates. 

(6.)  Obtain,  by  the  general  method,  the  equations  of  tangent 
and  normal  at  jP^  to  the  circle 

{x-x,y  +  (y-y^)^  =  a^ 

Ans.  < 

[  x^  -  x^^ 

(7.)  Show  that  the  same  equations,  obtained  by  transforma- 
tion of  coordinates  from  (10)  and  (11),  are 

{x,  -  x^)  (x  -  x^)  +  (j'l  -  Jc)  (y  -  Jc)  =  a\ 
and 

(j'l  -  j'c)  X  -  (-^1  -  -^c)  y  -  -^c  J'l  +  ^1  yc  =  o. 

(8.)  Reduce  the  results  of  the  last  two  examples  to  identical 
forms. 

(9.)  Find  the  equations  of  the  tangent  and  normal  to  the 
circle 

(x  -  2)2+  (y  _  sY  =  10. 
at  a  point  (5,  4). 


Ans. 


;^  X  +  y  -  ig  =  o, 

^  -  sy  +  7  =  Q- 


THE    CIRCLE.  85 

(10.)   Obtain,  by  the   general   method,  the   equation   of  the 
tangent  at  F^  to  the  circle 

x^-Yy^^-ax-^by-Vc^o. 

Ans.  x^x  +  y,y  +  ~{x  +  x)  +  -(y  +  y,)  +  c  =  o. 
2  2 

(11.)   Find    the   equations   of  the  tangent  and  normal  at  the 

origin  to  the  circle 

x"^  +  y^  —  7,  x  —  2  y  =  o. 


2  X  —  2>y  —  o. 

(12.)  The  projection  on  the  axis  of  x  of  that  part  of  the  tan- 
gent which  is  contained  between  (9Xand  the  point  of  tangency 
is  called  the  sub-tangent ;  in  like  manner  the  projection  of  that 
part  of  the  normal  which  lies  between  its  points  of  meeting  with 
the  curve  and  O  X'\s  called  the  sub-nor7nal. 

Find  the  lengths  of  the  sub-tangent  and  sub-normal  for  the 

circle 

x^  ^  ^2  _  ^2^ 

Suggestion.  It  follows  from  the  definition  that  the  sub-tan- 
gent equals  the  intercept  of  the  tangent  on  O  X,  diminished  by 
the  abscissa  of  the  point  of  tangency. 

Ans.    — ,  X,. 

(13.)  Prove  that  the  two  circles  given  in  §  45,  Ex.  11,  have  a 
common  tangent  at  the  point  (8,  3)  ;  show  that  the  line  which 
joins  their  centres  is  perpendicular  to  this  tangent  and  passes 
through  the  point  of  tangency  of  the  circles. 

(14.)  What  must  be  the  relation  between  a^,  b^,  and  a  in  order 
that  the  line 


may  touch  the  circle 


X      y 
x"  +y^  =  a"} 


Afis.   a  = 


V^i'  4-  V 


86  ANALYTIC    GEOMETRY. 

(15.)    When  will  the  line 

y  z=  X  X  -\-  b 
touch  the  circle 

Ans.    When  ^2  =  ^2  (j  _j.  ;^2)^ 

(16.)    When  will  the  line 

X  cos  a  +  J'  sin  a  =/ 

be  tangent  to  the  circle 

^2  _^  ^2  _  ^2^ . 

Ans.    When/  =  ^. 

50.    The  equation  of  the  tangent  to  a  circle  from  a  given  poirtS, 
without  the  curve  may  be  found  as  follows  :  — 
Let  P^  be  the  given  point,  and 

J^2     _J.    j;2    _    ^2 ^jj 

the   equation   of    the   circle  ;    let  P'  be   the  unknown  point  ol 
tangency. 

By  §  49,  we  may  write  the  equation  of  the  required  tangent 

x'  X  +  yy  =  a% (2} 

where  x'  and  y'  are  at  present  unknown. 

Since  the  locus  of  (2)  must,  by  the  conditions  of  the  problem, 
contain  the  point  P^,  the  coordinates  of  P^  must  satisfy  (2)  , 
therefore 

x'  X,  +  y  y,^a^ =  .   (3} 

Again,  P'  is  by  supposition  a  point  of  the  circle,  and,  there 
fore,  its  coordinates  satisfy  (i),  and 

x'-^+y'^  =  a^ (4} 

We  have  now  two  equations,  (3)  and  (4),  connecting  b} 
means  of  the  known  quantities,  x^,  y^,  and  a,  the  unknown  quan 
ties  x'  and  y' ;  consequently  we  can  find  both  x'  and  j''  by  elimi- 
nating between  (3)  and  (4). 

It  is  evident  that  there  will  be  two,  and  only  two,  pairs  ol 


1 


THE    CIRCl  E.  87 

values  for  x'  and  y,  because  (3)  is  of  the  first,  and  (4)  of  the 
second,  degree.  From  this  we  infer  that  two  tangents  can  be 
drawn  to  the  circle  from  a  point  without,  a  fact  w'ell  known  in 
Geometry.  If  we  call  the  two  pairs  of  values  mentioned  above 
{x',  y')  and  (x",  y"),  we  shall  have,  by  substituting  them  in  turn 
in  (2),  the  equations  of  the  required  tangents 

x'  x  +  y'  y  =  a'^, 

x"  X  -^  y"  y  =  a% 
touching  the  circle  at  jP'  and  J^",  respectively. 

Note,  In  this  discussion  we  have  supposed  J^^  a  point  with- 
out the  circle,  because  we  know  from  Geometry  that  no  tangent 
can  be  drawn  to  a  circle  from  a  point  within.  If  we  apply  the 
same  method  when  /^j  lies  within  the  circle,  the  coordinates  of 
/"  and  P"  will  have  imaginary  values,  showing  that  there  are 
no  real  tangents  in  this  case.  If  J^^  lies  on  the  circle,  and  the 
same  method  is  used,  the  values  of  the  coordinates  of  J^'  and  J^" 
will  be  alike  and  equal  to  {x\,  y^)  ;  the  two  tangents,  therefore, 
will  coincide  in  this  case,  a  result  which  Geometry  also  teaches. 

EXAMPLES, 
(i.)    Find  the  equations  of  the  tangents  from  the  point  (7,  i) 
to  the  circle 

Here, 

^1  =  7,  yi  =  I,  ^'  =  25. 

Making  these  substitutions  in  equations  (3)  and  (4),  we  have 

7^'  +y  =  25, 


and 


x'^  +y^  =  25. 


From  these  equations  we  find  that  either 

x'  =  3  and  y'  =  4, 
or 

x^  =  4.  and  y  =  —  3, 


55  ANALYTIC    GEOMETRY. 

The  required  equations  are,  therefore, 

3  ^  +  47  =  25, 
and 

4^-37  =  25. 

(2.)  Find  the  equations  of  the  tangents  from  (—7,  —  i)  to  the 
circle 

x^  -l-y  =  25. 

C  4-^  -37  +  25  =  o. 
(3.)    What  are  the  equations  of  the  tangents  to  the  same  circle 

from  the  point  (  —  ,  — )  ?     Prove  that  the  lengths  of  these  tan- 
gents are  equal. 

(4.)  Find  the  equation  of  the  chord  which  joins  the  points 
where  tangents  from  P^  touch  the  circle 

This  may  be  done  by  substituting  the  coordinates  of  /"  and 
P^^  in  the  equation  of  a  line  which  passes  through  two  points,  or 
by  the  following  instructive  method  :  — 

The  equations  of  the  tangents  to  the  circle  at  P^  and  P^^  are 

x^  X  -\-  y'  y  =  a"^ 


and 


x"  X  +  y"  y  =z  a^. 


Since  by  supposition  P^  lies  on  each  of  these  lines,  we  have 
and 


x^  Xi  +  y^  _)',  r=  a^ 


x'^  Xj  +  y"  J,  =  a^. 


These  last  equations  may  also  be  regarded  as  the  equations  of 
condition  which  make  the  line 

x^x  +y^y^a^ 


THli    ClRCLt;. 


i>9 


pass  through  the  points  {x',y')  and  {x",y").     But  we  are  seek- 
ing the  line  joining  F'  and  P" ;  therefore 

is  the  required  equation. 

Remark.  If  F^  lies  on  the  circle,  this  chord  becomes  the 
tangent  to  the  circle  at  F^,  as  is  evident  from  its  equation. 

51.  A  diameter  of  any  curve  is  the  locus  of  the  middle  points 
of  a  set  of  parallel  chords. 

We  will  find  the  equation  of  a  diameter  of  the  circle 

^2  +  y2  ^  ^2 ^^^ 

Let  the  inclination  of  each  of  the  parallel  chords  be  y^,  and  let 
their  slope  be  A^.  The  equations  of  these  chords  may  then  be 
written :  — 

y  ^\x  ^b^    .  .   (i) 
y  =  \x  +  d,    .  .  (2) 
y  =  X^x-^b,    .  .   (3) 
etc. 

In  the  figure  the  line  (i) 
represents  the  locus  of  the 
first  of  these  equations  ;  the 
extremities  of  this  chord  are 
F^'  and  F^",  and  its  middle 
point  is  F'.  In  like  manner, 
F.J  FJ',  etc.,  are  other  chords 
having  the  equations  (2),  etc. 

By  §  II,  corollary,  the  coordinates  of  F'  are 


,/  _ 


,y_y'+y''. 


Now/'/ and  7^/' are  the  points  where  the  chord  (i)  cuts  the 
circle  ;  therefore,  by  §  20,  their  coordinates  may  be  found  by 
eliminating  between  equations  (o)  and  (i). 


9©  ANALYTIC    GEOMETRY. 

Squaring  (i)  and  substituting  in  (o),  we  have 

x^  +  Xj^  x^  +  2  l?i  X^  X  +  l^j^  =  a% 
or 

(I  +  A/-)  X"  +  2  /^i  Xj  jx:  +  (<^j^  —  <z^)  =  o, 

and  the  two  roots  of  tliis  equation   are  x/  and  x/'.     The  sum 
of  the  roots   of  such   an   equation   I's,  mi ?ius  the  coefficient  of  x 
divided  by  the  coefficient  of  x\  * 
Therefore 


X-y    -\-  X-^  0-,  Aj 

2  ~  ~   I  +  A, 


^.  =  i^^I^  =  _  JlL^^ (^) 


Substituting  this  value  in  (i)  and  solving  for/,  we  shall  obtain 
y' ;  we  have 

f  ^^  ^i'  7 


reducing  to  a  common  denominator  and  combining  we  obtain 

■^    I  +  V 

In  the  same  way  the  coordinates  of  the  middle  point  of  the 
second  chord  may  be  found  to  be 

^2  K  ^  ^o 


-;,    and 


i+A/^'    "--     I  +A,2. 

and  so  for  the  middle  point  of  any  other  chord. 

Now  the  equation  of  any  locus  must  express  an  invariable 
relation  existing  between  the  coordinates  of  all  points  on  that 
locus.     The  coordinates  of  the  different  middle  points  only  differ 

*  In  the  equation  a.i''  -\-  ^  x  -^  •>/  =  o  the  roots  are 


-  ;3  ±    |-)3-^  -4^7 
2  a 

a 

Accordingly  the  sum  of  the  roots  is  —  — ,  and   (a  fact  which  we  add  for  future  refer- 


TIIF.    CIRf'T^E. 


91 


in  the  (luantities  /^i,  A.,  etc.,  whicli  enter  as  factors  into  the  vahies 
of  each  coordinate  ;  dividing  y'  by  x  we  obtain 

y      I 

S  =  -V     <5) 

a  rehition  which  is  independent  of  b^,  and  consequently  must  be 
true  where  P'  represents  the  middle  point  of  the  chord  (2),  or 
(3),  or  any  other  of  the  system  of  parallel  chords. 

liquation  (5)  expresses  an  invariable  relation  between  the 
coorcHnates  of  each  point  of  the  required  locus,  and  dropping 
the  accents  we  have 

i=-i  °'"  y=-\''^ (^) 

tile  equation  of  the  diameter  which  bisects  all  chords  which  have 
a  slope  A,. 

Since  equation  (6)  is  of  the  first  degree^  its  locus  must  be  a 
strai<;iit  line  ;  as  there  is  710  constant  tcrtn^  the  origin  is  a  point  of 
tile  locus. 

Equation  (6)  may  be  written 

y^^x (7) 

where 

^  =  -f (8) 

By  §25,  RemaT.v,  equation  (7)  represents  a  line  through  the 
origin  with  a  slope  A;  and,  by  §33,  Cor.  2,  and  equation  (8), 
this  line  must  be  perpendicular  to  the  chords  which  it  bisects. 

NoiK.  'I'liese  properties  are  well-known  pro|)erties  of  the 
(liatneter  of  the  circle,  and  the  equation  of  the  diameter  might 
have  been  easily  found  by  means  of  them  ;  but  we  have  used  the 
general  method  of  finding  the  diameter  of  any  curve,  in  order  to 
illustrate  it  by  a  simple  example. 


92 


ANALYTIC    GEOMETRY. 


52.  We  may  readily  show  that  every  chord  through  the  centre 
of  a  circle  is  a  diameter;  for  by  §  25,  Remark,  any  such  chord 
may  have  its  equation  written  in  the  form 

y  =  ^x (i) 

Now,  whatever  the  value  of  A,  we  can  find  a  quantity  X^,  such  that 

K  =  -{,     or    A=-^. 

Since  any  quantity,  from  00  to  —00,  may  be  the  slope  of  a 
line,  it  is  always  possible  to  draw  a  set  of  chords  which  shall 
have  the  value  of  Aj,  thus  found,  for  their  common  slope ;  then, 
by  §51,  these  chords  will  be  bisected  by  the  locus  of  equation 
(i),  and  that  locus  must  be  a  dia?neter  of  the  circle. 

EXAMPLES. 

(i.)    A  system  of  straight  lines  parallel  to 

2x  -^Ty  =  S 


intersects  the  circle 


x^  +  /  =  40- 


Find  the  equation  of  the  line  which  bisects  the  chords  thus 

formed. 

2 
Here  Aj,   the  slope  of  the  system  of  chords,  is ;   conse- 

7  ^ 

quently  A,  the  slope  of  the  diameter,  must  be  -,  and  the  equation 

of  the  diameter  is 

7 
y  =  -x, 

or 

'J  X  —  2  y  z^  o. 

(2.)    Show,  by  transforming  coordinates,  that  the  equation  of 
the  diameter  of  the  circle 

{x-x,Y  ^  {y-y,Y  =  a\ 


THE    CIRCLE.  93 

which  bisects  all  chords  with  a  slope  A,,  is 

y  —  jc  =  ~  Y  ^^  ~  ■^''•^* 

(3.)   Find  the  equations  of  the  diameters  of  the  circles 
x'  +/  —  4.r  +  4>'  —  I  —  o, 
.r'  -f-/4-6.T  -  3J^-  I  =0, 

which  bisect  all  chords  with  an  inclination  of  135°. 

(4.)  Show  that  the  extremities  of  a  diameter  of  a  circle  whose 
centre  is  the  origin  are  the  points 

(a  a  X     \     /  a  a\      \ 

(5.)  Prove  that  the  lines  joining  the  extremities  of  two  diam- 
eters of  a  circle  are  parallel. 

(6.)  Prove  that  the  tangents  to  a  circle  at  opposite  extremi- 
ties of  any  diameter  are  parallel. 

(7.)    Show  that  the  extremities  of  a  diameter  of  the  circle 

{x  -x^)^  -^r  {y  -y,f=r'' 
are  the  points 


l^c  ±  ■  ,.  y.  ±  -j=T.X 

^        V 1  +  /^  V I  +  ^^ 


(8.)  If  through  one  of  the  points  of  intersection  of  two  circles 
a  diameter  of  each  circle  is  drawn,  the  straight  line  which  joins 
the  extremities  of  these  diameters  passes  through  the  other  point 
of  intersection,  and  is  parallel  to  the  line  joining  their  centres. 
Prove  this  for  the  circles  given  in  §  45,  Ex.  9. 

53.  We  will  close  this  chapter  by  finding  the  polar  equation 
of  the  circle. 

Let  Ox  be  the  initial  line,  and   O  the  pole.     Let  C  be  the 


94 


ANALYTIC    GEOMETRY. 


centre   of   the  circle 

with  coordinates  (r^, 
i'>e),  and  let  a  be  the 
radius.  The  coordi- 
nates of  P,  any  point 
of     the     curve,     are 

In  the  triangle 
O  C  P,  we  have  by 
Trigonometry 


Now 


C  P'-^  O  C'-V  OP'-  2  OCX  OPx  cos  COP.  .  (i) 


OP^r. 


CP=a,    OC=r, 

From  the  figure  and  §  3,  we  have 

C  O  P=  XOP- XOC 

=  {y  -  x%. 

Substituting  these  values  in  (i),  we  obtain 

a^  =  7-^  _|_  r^  —  2  ^c  r  cos  {&  —  &q\ 

which  must  be  the  required  equation. 
This  may  be  written  in  the  form 

r'  —  2  r^  r  cos  (i>  —  t>c)  +  ^'^  —  a^  =  o. 

It  is  evident  from  the  form  of  this  equation  that  with  each 
value  of  ft  there  will  be  two  values  for  r ;  a  fact  which  agrees  with 
the  geometric  properties  of  the  circle. 

Corollary  1.  If  C  is  on  OX,  0^  =  o,  and  the  equation 
becomes 

r^  —  2  7\  r  cos  i>  +  7\'  —  a'^  =  o. 

Corollary  2.  When  the  pole  is  on  the  circumference,  and 
the  initial  line  passes  through  the  centre, 


THE    CIRCLE.  95 

r^  —  a  and  ^^  —  o\ 
in  this  case  the  equation  reduces  to  the  form 
r  —  2  a  cos  0-. 

This  equation  gives  but  one  value  of  r  for  each  value  of  ^ ; 
as  we  have  divided  each  term  by  r,  however,  r  may  equal  o  with 
each  value  of  ^,  a  fact  which  is  apparent  from  the  hypothesis. 

Corollary  3.  If  the  centre  is  the  pole,  r^.  =  o,  and  the  equa- 
tion becomes 

r  =  ±  a^ 
as  shown  in  §  38. 

EXAMPLES, 
(i.)    Show  that  the  polar  equation  of  the  circle,  the  origin 
being  on  the  circumference  and  the  initial  line  a  tangent,  is 

r  =:  2  ^  sin  i9-. 

(2.)  Obtain  the  equation  of  Cor.  3  from  the  rectangular  equa- 
tion referred  to  the  centre,  by  transformation  of  coordinates. 

(3.)    Obtain  the  equation  of  Cor.  2  from  that  of  §  45,  Cor.  2. 

(4.)  Obtain  the  general  form  of  the  polar  equation  from  the 
rectangular  equation 

(5.)  Prove  that  the  two  values  of  r,  given  by  the  general  polar 
equation,  are  real  and  different,  real  and  equal,  or  imaginary, 

according  as  sin  (&  —  d-^)  is  less  than  —  ,  equal  to  — ,  or  greater 

than    — ,  in  numerical  value.     Show  that  these  results  mav  also 

be  obtained  from  the  figure  by  means  of  well-known  properties 
of  the  circle. 

(6.)  Prove  the  theorem  of  §  45,  Ex.  13,  using  polar  coordi- 
nates. 


CHAPTER  VI. 
LOCI. 

54.  In  finding  the  equation  of  the  circle,  we  expressed  in  an 
equation  its  most  familiar  property.  But  the  circle  may  also  be 
defined  by  any  other  essential  property,  and  we  can  find  its 
equation  by  expressing  that  essential  property  in  an  equation 
connecting  the  coordinates  of  any  point  of  the  curve. 

In  §  51,  a  diameter  of  a  circle  is  defined  as  the  locus  of  the 
middle  points  of  a  set  of  parallel  chords,  — a  definition  different 
from  the  familiar  one  of  Geometry,  but  expressing  a  well-known 
property  of  diameters.  From  this  new  definition,  however,  we 
found  the  invariable  relation  which  exists  between  the  coordi- 
nates of  every  point  of  the  locus,  and  we  obtained  the  equation 
of  a  diameter  by  expressing  this  invariable  relation  in  an  equa- 
tion. The  result  was  recognized  as  representing  a  straight 
line  through  the  centre,  and  in  this  way  we  obtained  from  the 
equation  the  properties  by  which  a  diameter  is  commonly 
defined.  If  these  properties  were  unknown  to  us,  we  should  be 
no  less  certain  of  their  truth. 

In  like  manner,  if  a  point  moves  according  to  a  given  geo- 
metric law,  we  can  find  the  equation  of  its  path,  or  locus,  though 
ignorant  of  the  form  of  that  path.  This  equation  may  be  recog- 
nized as  the  equation  of  some  known  curve,  in  which  case  we 
can  determine  at  once  the  locus  of  the  moving  point ;  if,  how- 
ever, the  equation  is  not  familiar,  we  can  deduce  from  it  the 
form  and  properties  of  the  curve. 


97 


In  this  chapter  we  shall  apply  this  method  to  several  loci 
defined  by  properties  more  complicated  than  those  mentioned 
above,  and  we  shall  add  a  number  of  other  examples  leaving  the 
work  wholly  or  in  part  to  the  student. 

55.  Let  us  find  the  locus  of  the  vertex  of  a  triangle  when  the 
base  and  the  difference  of  the  squares  of  the  sides  are  given. 

Take  the  base  for  the  axis 
of  X,  and  its  left-hand  ex- 
tremity for  the  origin.  Call 
the  vertex  /^,  with  coordi- 
nates, X  =  O  M,  y  =  M  P ; 
let  the  length  of  the  base, 
O  £>,  be  r,  and  let  m-  repre- 
sent the  constant  difference 
between  the  squares  of  the 
sides. 

The  definition  of  the  locus 
may  be  expressed  by  the  equation 

O^'  -  nT^  =pi\ (i) 

which  will  be  the  equation  of  the  required  locus  when  we  express 
O  P  and  D  P\x\  terms  of  .r,  r,  and  c. 

Since  the  coordinates  of  D  are  (<^,  o),  we  may  write,  by  §  10, 

'OP^  =  x'  +  y\ 

Wp''  =  {c^xY  +  y\ 

Substituting  these  values  in  equation  (i),  we  have 

.v-^+/_[(r-.r)2+/]  ^m\ 

which  may  be  reduced  to  the  form 


2  c  X  —  r  —  m' 


or 


X  — 


c^  +  tn^ 


UMfVrrFcSlTY  OF  G,^LJFCV^NIA 
-ARTMENT  O-  CIVIL  Ei-^G.:,-;.. 


98 


ANALYTIC    GEOMETRY. 


From  the  form  of  this  equation,  we  know  that  the  locus  is  a 
straight  line  parallel  to  O  F,  and  therefore  perpendicular  to  the 


base  of  the  triangle,  at  a  distance  from  O  equal  to 


<:^  +  ni^ 


2  c 


It  should  be  noticed  that  the  other  segment  of  the  base  is 
,  a  result  which  may  also  be  obtained  by  taking  the  origin 


2  c 
at  D. 

(2.)  Show  that  if  the  sum  of  the  squares  of  the  sides  is  given 
instead  of  their  difference,  the  locus  is  a  circle  with  the  centre  at 
the  middle  of  the  base.  Find  the  length  of  the  radius  of  this 
circle. 

(3.)  Solve  problems  (i)  and  (2),  taking  the  origin  at  the 
middle  of  the  base  (v.  Ex.  5,  Fig.),  and  show  that  the  results 
agree  with  those  already  found. 

(4.)  Given  the  base  of  an  isosceles  triangle,  find  the  locus  of 
the  vertex. 

Ans.   A  straight  line  perpendicular  to  the  base  and  bisect- 


(5.)     Given  the  base  and  the  ratio  of  the  sides  of  a  triangle, 
find  the  locus  of  the  vertex. 

Let  the  base  be  the  axis  ^ 

of  X,  and  its  middle  point 
the  origin.  Let  the  length 
of  the  base  equal  2  c,  and 
the  ratio  of  C  P  Xo  DP 
he  m  :  n. 

With  this  notation,  the 
coordinates  C  and  Z>  are 
( —  r,  o)  and  (c,  o) ;  the 
vertex  iPhas  coordinates  {x,y).     We  have,  by  §  10, 

CP 


Z>P=  ^{x-cf  ^y\ 


LOCI.  99 

The  equation  of  the  required  locus  may,  therefore,  be  written 


squaring  and  clearing  of  fractions,  this  becomes 

nr  \{x  -  c-y-  +/]  =  71"  \{x  +  cY  +/], 
or 

(w-  -  if)  x^  -  2  {7n^  +  it^)  c  X  -{■  {m^  —  ti^)  y  +  {nv^  —  rF)  c^  =  o, 
which  may  be  written 

x^  -  2  —^ 2  ^  ^  +  y^  +  r  z=  o. 


This  may  be  readily  shown  to  be  the  equation  of  a  circle,  with 
^        -,  c,  o  I   and  its  radius  equal  to 


2  m  n 
m^  —  n^ 


(6.)  Given  the  base  of  a  triangle,  and  m  times  the  square  of 
one  side  plus  n  times  the  square  of  the  other ;  find  the  locus  of 
the  vertex. 

Afis.    Using  the  same  axes  and  notation  as  in  the  last  example, 

.  ,    .  /m  —  ft        \ 

the  locus  IS  a  circle  with  its  centre  at  i c,  o  i. 

\m  +  11        J 

(7.)  Given  the  base  and  sum  of  the  sides  of  a  triangle;  find 
the  equation  of  the  locus  of  the  vertex. 

Ans.  If  2  <r  is  the  base  and  2  a  the  sum  of  the  varying  sides, 
and  the  axes  as  in  Ex.  5,  we  have  for  the  equation  of  the  locus 


V(^  +  (^f  +/  +  \^{x  -  cj^  +  f  =  2  a. 
Removing  one  radical  to  the  second  member,  and  squaring 
twice,  we  shall  have  for  the  required  equation 

{a"  -  r)  x"  +  ^2/  =  a^  (^2  _  ^ly 

This  is  neither  the  equation  of  a  straight  line  nor  of  a  circle ; 
we  shall,  however,  have  occasion  to  study  its  locus  hereafter. 


lOO  ANALYTIC    GEOMETRY. 

(8.)  Show  that  if  the  difference  of  the  sides  in  the  last  exam- 
ple is  given  equal  to  2  a,  instead  of  their  sum,  the  equation  of 
the  locus  has  the  same  form. 

(9.)  Find  the  locus  of  a  point  such  that,  if  straight  lines  be 
drawn  to  it  from  the  four  corners  of  a  given  square,  the  sum  of 
their  squares  is  constant. 

Ajis.  a  circle  whose  centre  is  the  centre  of  the  square,  and 
whose  radius  is  V2  {>n^  —  ^^),  where  each  side  of  the  square  is 
2  a  and  the  given  constant  is  represented  by  8  nt^. 

(10.)  Find  the  locus  of  a  point  whose  distances  from  two 
straight  lines,  given  by  their  equations,  have  a  given  ratio,  m  :  n. 

Let  the  equations  of  the  lines  be 

Ax-^By^C—  o, (i) 

A^ x-\.  B^ y  +  C'^o; (2) 

and  let  P^  with  coordinates  {x\y^)  be  the  moving  point. 
The  distances  of  P^  from  lines  (i)  and  (2)  are,  by  §  36, 

Ax^  ^  By^  ^  C         ^     A'  x'  +  B'  y'  +  C 
- —      and     — 

V^'  +  £'  Va'-'  +  ^" 

Putting  m  times  one  of  these  equal  to  n  times  the  other,  we 
have  the  equation  of  the  locus,  where  x'  andy  are  the  variable 
coordinates.  Clearing  of  fractions  and  writing  x  for  x'  and  y 
for  y',  we  evidently  have  an  equation  of  the  Jirsf  degree;  there- 
fore the  required  locus  is  a  straight  line. 

(11.)  Show  that  the  equations  of  the  bisectors  of  the  angles 
between  the  lines 

A X  -\-  By  +  C=  o     and     A^ x  ^  B^ y  ^  C  =  o 
are 

Ax  +  By+C  ^      A'x  +  B'y+a 

VA^  +  B'  V'A'^'^B"'   ' 

(12.)  A  point  moves  so  that  the  sum  of  the  squares  of  its 
distances  from   the  four  sides   of  a  given  square  is  constant ; 


LOCI. 


I C  T. 


show  that  the  locus  of  the  point  is  a  circle  ;  find  its ,  ':eritre,  ,and> 

radius.  ^  ^' '  >  J . .  >  ^   ' ,  >' ;  *-.',  ,^ '.  \ 

(13.)    Find  the  locus  of  a  point,  the  square  of  whose  distance 

from  a  given  point  is  proportional  to  its  distance  from  a  given 

line. 

A^is.    A  circle. 

Suggestion.  Take  the  fixed  line  for  the  axis  of  y,  and  the 
perpendicular  upon  it  from  the  fixed  point  for  the  axis  of  x. 

(14.)  Find  the  equation  of  the  locus  of  a  point  whose  dis- 
tance from  a  fixed  point  always  equals  its  distance  from  a  fixed 
line. 

Ans.  y  =  2  Xj^x  —  x^  [axes  as  in  13]. 

56.  There  are  many  problems  in  whfch  it  is  difficult  to  express 
at  once  the  given  conditions  in  terms  of  the  coordinates  of  the 
moving  point  and  known  quantities.  In  such  cases  we  may 
introduce  other  variable  quantities,  and  then  find  enough  equa- 
tions expressing  necessary  relations  of  the  parts  of  the  figure  to 
enable  us  to  eliminate  the  variable  quantities  thus  introduced. 
We  shall  then  have  an  equation  which  fulfils  the  requirements  of 
the  definition  of  the  equation  of  a  curve. 

(i)  To  illustrate,  let  us  find  the 
locus  of  the  middle  points  of 
rectangles  inscribed  in  a  given 
triangle. 

We  will  take  the  base  and 
perpendicular  from  the  vertex  as 
axes,  and  will  suppose  the  tri- 
angle given  by  means  of  the  in- 
"x  tercepts  which  the  sides  cut  from 
these  axes. 


/ 

Y 

F 

/ 

\ 

F 

/ 

-.p/'"' 
"'["^^-. 

\ 

\ 

A      I 

D 

0  M             ( 

a 

B        X 

Let 


OA=a,     OB 


OC  =  b. 


102  ANALYTIC    GEOMETRY. 

Let  I)  E,  F  G  represent  any  rectangle  inscribed  in  the  tri- 
angle,, and  iec  its  middle  point  be  P. 

We  can  easily  express  the  coordinates  of  P  in  terms  of  those 
of  E  and  F;  for,  from  the  figure  and  §  ii,  it  is  evident  that  x 
must  equal  the  half  sum  of  x^^  and  x^  (the  abscissas  of  E  and 
F),  and  y^  one-half  the  common  ordinate  of  these  points.  Call 
this  common  ordinate  k;  then  the  equation  of  ^  i^  may  be 
written 

y-^k (i) 

By  §  22,  the  equations  of  ^C  and  B  C  are 


and 


-  +  i=I, (2) 

a       P 


^-f=' (3) 


Since  E  and  F  are  the  points  where  the  line  (i)  meets  the 
lines  (2)  and  (3),  we  have  by  §  20 

therefore 

x^  +  Xp      (a  -f.  a')  (b  -k) 

•^  =  ~~2~~  =  Vb  ^ ^4^ 

also  for  the  point  P 

y  =  i (5) 

Equations  (4)  and  (5)  contain  the  coordinates  of  P,  the  vari- 
able k,  and  the  known  constants  a,  a^  and  b.  We  may,  there- 
fore, by  combining  these  equations  eliminate  k  and  obtain  an 
equation  connecting  x  and  y  by  means  of  the  given  quantities 
^,  a\  and  b.     This  equation  is 

_  (^  +  a')  {b  -  2  y^ 


LOCI.  103 

or 

2  X        2y 

the  equation  of  a  straight  line  bisecting  the  base  and  perpendic- 
ular of  the  triangle,  for  its  intercepts  are 

a  +  a^  ,       b 

— ^ and 


Remark.  The  student  will  remember  that,  in  finding  the 
equation  of  a  diameter  of  a  circle,  we  first  expressed  the  coor- 
dinates of  any  point  of  the  locus  in  terms  of  another  variable, 
the  intercept  of  any  chord  on  O  Y.  That  problem,  therefore,  is 
one  of  the  class  we  are  now  considering. 

(2.)  A  line  is  drawn  parallel  to  the  base  of  a  given  triangle 
and  its  extremities  joined  transversely  to  those  of  the  base ;  find 
the  locus  of  the  point  of  intersection  of  the  joining  lines. 

With  the  same  axes  and  notation  as  in  the  last  example,  the 
equations  of  the  joining  lines  may  be  formed  by  §  27.     Finding 
the  coordinates  of  the  point  where  these  lines  meet,  we  have, 
after  eliminating  k^  the  equation  of  the  locus, 
2  X       y 

which  represents  the  medial  line  drawn  from  the  vertex  of  the 
triangle. 

(3.)  A  line  of  constant  length  moves  with  its  extremities  on 
two  strai2:ht  lines  at  right  angles  to  each  other  ;  find  the  locus  of 
its  middle  point. 

This  may  be  done  conveniently  by  introducing  as  a  third  vari- 
able the  angle  which  the  moving  line  makes  with  one  of  the 
fixed  lines. 

Ans.    A  circle.     Compare  §11,  Ex.  9. 

(4.)  A  square  is  moved  so  as  always  to  have  the  two  extrem- 
ities of  one  of  its  diagonals  upon  two  fixed  straight  lines  at  right 


I04  ANALYTIC    GEOMETRY. 

angles  to  each  other  ;  show  that  the  extremities  of  the  other 
diagonal  will  at  the  same  time  move  upon  two  other  fixed 
straight  lines  at  right  angles  to  each  other. 

(5.)  From  one  extremity  A'  oi  a  fixed  diameter  ^'^  of  a 
given  circle,  a  secant  is  drawn  through  any  point  P'  of  the  cir- 
cumference ;  B.t  I"  3.  tangent  is  drawn  to  the  circle,  and  a  perpen- 
dicular to  the  tangent  is  drawn  from  A,  and  extended  to  meet 
the  secant  in  J^.     Find  the  locus  of  jP. 

Take  the  fixed  diameter  as  the  axis  of  x,  and  the  centre  as 
origin.  We  shall  use  as  auxiliary  variables  the  coordinates  of  J^', 
The  points  A'  and  A  have  coordinates  (-  a,  o)  and  (a,  o). 

We  can  now  write  the  equation  of  A'  J^'  by  §  27,  and  oi  A  F 
by  §  2)2>^  Cor.  2  ;  for  we  know  the  slope  of  the  tangent  at  F'  by 
§  49.  Since  F  is  the  point  of  intersection  of  A'  F  and  A  F,  its 
coordinates  may  be  found  by  §  20,  and  will  be 

X  —  2  x'  -If-  a,    y  =  2  y, 

Expressing  x^  and  y  in  terms  of  x,  y,  and  a,  by  means  of  these 
equations,  and  substituting  the  values  thus  found  in  the  equation 
of  condition  which  puts  F'  upon  the  circle, 

x^""  +y^  =  a\ 

we  have  for  the  equation  of  the  locus 

{x  -  ay  +y'^  =  4a% 

which  represents  a  circle  with  its  centre  at  A,  and  A  A^  for  its 
radius. 

(6.)  Given  the  base  of  a  triangle  and  the  length  of  the  medial 
line  drawn  from  one  of  its  extremities ;  find  the  locus  of  the 
vertex. 

Take  the  base  as  the  axis  of  x,  and  the  end  from  which  the 
medial  line  is  drawn  as  the  origin.  Introduce  as  auxiliary  vari- 
ables the  coordinates  of  the  other  extremity  F^  of  the  given 
medial  line. 


LOCI.  105 

Ans.  A  circle  with  centre  at  the  point  (—  c,  o)  and  radius  2  w, 
where  c  and  m  represent  the  lengths  of  the  base  and  medial 
line. 

(7.)  To  find  the  locus  of  the  centre  of  a  circle  which  has  a 
given  radius  and  passes  through  a  given  point. 

Let  F'  be  the  unknown  centre.  The  equation  of  the  circle 
may  then  be  written 

{x-x'Y  +  {y-yy  =  a\ 

Making  this  pass  through  the  given  point  F^,  we  have 

changing  the  signs  of  the  quantities  in  the  parentheses  (but  not 
of  their  squares)  and  writing  x  and  y  for  the  variables  x'  and  y', 
this  equation  becomes 

(x-x,y  +  (y  -y,y  =  a\ 

the  equation  of  a  circle  with  P^  for  the  centre,  and  a  for  its 
radius. 

Remark.  This  equation  may  be  easily  obtained  from  the 
fact  that  the  distance  of  the  moving  centre  from  the  fixed  point 
must  be  equal  to  the  given  radius ;  but  the  method  used  above 
is  instructive. 

(8.)  Find  the  locus  of  the  middle  points  of  chords  of  a  given 
circle,  drawn  from  a  fixed  point  on  the  circumference. 

Ans.  A  circle  with  the  radius  of  the  given  circle  for  its 
diameter. 

Compare  with  §  45,  Ex.  13. 

57.  Instead  of  expressing  in  an  equation  the  geometrical 
definition  of  the  locus,  and  then  substituting  for  each  quantity  in 
this  equation  its  value  in  terms  of  the  coordinates  of  the  moving 
point  and  known  quantities,  it  is  often  more  convenient  to  use 
as  the  primary  equation  one  which  expresses  some  necessary 
relation  of  the  parts  of  the  figure  other  than  the  one  explicitly 


io6 


ANALYTIC    GEOMETRY. 


given.  From  our  hypothesis  we  may  be  able  to  express  all 
quantities  in  this  equation  in  terms  of  the  variable  coordinates 
and  known  quantities,  and  so  find  the  equation  of  the  locus. 

We  will  illustrate  by  an  example  in  which  this  method  is  used 
in  connection  with  that  of  §  56. 

(i.)  Given  the  base  and  sum  of  the  sides  of  a  triangle,  if  the 
perpendicular  be  produced  beyond  the  vertex  until  its  whole 
length  is  equal  to  one  of  the  sides,  find  the  locus  of  the  extrem- 
ity of  the  perpendicular. 

Take  the  axes  as  in  §  55, 
Ex.  I ;  let  the  base  be  r,  and 
the  sum  of  the  varying  sides 
be  m  J  let  M  F  ht  made 
equal  to  O  C,  whatever  the 
position  of  C. 

We  may,  therefore,  write 


OC  =  MF=y,  CD^m 
Now,  from  the  figure,  it  is  evident  that 


y- 


and 


MC"  =  CZ>^ 


(9  J/2    ^  ^2 

MD''  =  (m  -  yf 


(c  -  x)\ 


Putting  these  values  oi  M  C^  equal  to  each  other,  we  have  the 
expression  of  a  necessary  relation  between  x  and  j',  in  an  equa- 
tion which  contains  only  those  quantities  and  the  given  con- 
stants m  and  c.     Expanding  and  reducing,  this  equation  becomes 


2  c  X  —  2  7ny  +  m^ 


€"  = 


which  is  of  the  first  degree,  and  therefore  represents  a  straight 
line. 

(2.)    A  line  of  constant  length  moves  with  its  ends  on  two 
straight  lines  at  right  angles  to  each  other,  and  perpendiculars 


LOCI.  107 

to  the  lines  are  raised  at  its  extremities;  find  the  locus  of  their 
intersection. 

Ans.    A  circle. 

58.    Polar  coordinates  may  often  be  employed  to  advantage, 
(i.)    To  tind   the  locus  of  the  middle  points  of  chords  of  a 
circle,  which  pass  through  a  given  point. 

Let  the  given  point 

p    ^„^    "N^  be  the    pole,    and  the 

^,^^  X  line  joining  it  with  the 

centre    be    the    initial 
19 [ \ line.     Let    F'  F"    be 

O  \  C  j  X 

any  chord  which  passes 
through  (9,  and  let^Pbe 
\^  yr  its  middle   point,  with 

coordinates  /'  and  &. 
It  is  evident  from  the  figure  that  i>  will  be  the  same  ior  F,  F\ 
and  F^\  and  that 

^  =  —J-    (0 

The  equation  of  the  circle  may  be  written,  by  §  53,  Cor.  i, 

r-  —  2  ;-c  r  cos  i>  +  Tc^  —  <?-  =  o (2) 

When  />  equals  X  O  F",  the  roots  of  this  equation  are  r'  and 
r";  therefore,  by  equation  (i),* 

r  =  r^cos  d-, (3) 

and  similarly  with  every  value  of  i> ;  for  though  for  certain  val- 
ues of  l^,  r'  and  r"  are  imaginary,  yet  r,  their  half-sum,  is  always 
real  and  equal  to  r^  cos  />. 

Equation  (3)  is,  therefore,  the  equation  of  the  required 
locus,  which,  by  §  53,  Cor.  2,  must  be  a  circle  with  r^  for  a 
diameter. 

*  V.  foot  note  on  p.  90. 


lo8  ANALYTIC    GEOMETRY. 

(2.)  Find  the  locus  of  the  middle  points  of  chords  drawn 
from  the  extremity  of  any  diameter  of  a  circle. 

Compare  with  §  56,  Ex.  8. 

(3.)  Find  the  locus  of  a  point  which  moves  so  as  to  divide 
in  a  given  ratio  all  lines  joining  a  fixed  point  with  a  fixed 
straight  line. 

59.  A  great  many  problems  in  loci  lead  us  to  equations  of  an 
unfamiliar  form.  We  have  had  an  illustration  of  this  in  Ex. 
7,  on  page  99.  In  such  a  case  the  locus  will  usually,  as  in  the 
example  just  referred  to,  not  come  under  any  of  the  forms  of 
curves  we  have  so  far  studied.  There  is,  however,  one  method 
by  which  we  can  sometimes  determine  what  the  locus  of  an 
unfamiliar  equation  is. 

Consider,  for  example,  the  equation 

xy  =  0 (i) 

which  is  in  an  unfamiliar  form  since  it  is  of  the  second  degree 
and  yet  is  not  of  the  form  of  the  equation  of  a  circle  (§  46). 
It  is  clear  that  any  point  whose  abscissa  or  whose  ordinate  is 
zero  lies  on  the  curve  (i),  but  that  no  other  points  lie  on  it. 
The  locus  of  (i)  therefore  consists  of  two  parts  :  first  the  axis 
of  X,  second  the  axis  of  Y. 

As  a  second  example  let  us  consider  the  equation 

x'^  —  X  y  ■=  Q '  .  •  •  .  (2) 

This  equation  may  be  written 

x{x  -  y)  =  o ^  •  •   .  (3) 

It  will  therefore  be  satisfied  by  the  coordinates  of  any  point 
whose  abscissa  is  zero,  i.e.,  of  any  point  on  the  axis  of  Y. 
There  is,  however,  another  way  in  which  (3)  can  be  satisfied, 
namely  by   the  equality  o{  x  and>'.     The   locus   of   (2)  or  (3) 


LOCI.  lor 

therefore  consists  of  two  parts:   first   the  axis    of  K,  a:  =  o 
second  the  straight  line  x  —  y  =  o.     Moreover  it  is  clear  thai 
a  point  not  lying  on  either  of  these  two  lines  does  not   belon*; 
to  the  locus,  since  for  such  a  point  neither  x  nor  x  — jv  is  zero, 
and  therefore  (3)  is  not  satisfied. 

The  general  principle  of  which  the  two  foregoing  examples 
are  illustrations  may  be  stated  as  follows:  Suppose  that  all  the 
terms  of  the  equation  we  wish  to  investigate  have  been  trans- 
posed to  the  left-hand  side,  and  suppose  that  this  left-hand 
side  can  then  be  factored  into  two  factors  which  we  will  de- 
note by  the  letters  u  and  v.  The  equation  we  wish  to  investi- 
gate may  therefore  be  written 

uv  =  o (4) 

where  we  must  not  forget  that  the  letters  ?/  and  v  stand  for 
certain  more  or  less  complicated  expressions  involving  the 
variables  x  and  r.  Then  the  curve  (4)  consists  of  two  and  071/ y 
two  parts,  first  the  curve 

^  =  o (5) 


and  second  the  curve 


(6) 


Let  (^ijJi)  be  any  point  on  the  curve  (5)  ;  {x^,y^  any 
point  on  the  curve  (6)  ;  and  (^v,,  j,)  any  point  which  does  not 
lie  on  either  of  the  curves  (5)  or  (6).  The  statement  we  have 
made  will  be  proved  if  we  can  show  that  (a",,  J,)  and  {,x^^y^ 
both  lie  on  (4),  but  (jCg,  y^  does  not. 

In  order  to  find  out  whether  ix^^y^  lies  on  (4)  or  not,  we 
must  replace  the  variables  {x,)')  in  (4)  by  the  constant  values 
(;c,,>',)  and  then  see  whether  the  equation  is  fulfilled.  In  do- 
ing this  we  must  remember  that  u  is  merely  an  abbreviation  for 
a  certain  expression  in  the  variables  x  and  y.  This  expression 
will  therefore  take  on   a   certain  constant  value,  which  we  will 


110  ANALYTIC    GEOMETRY. 

call  u^,  when  these  variables  are  replaced  by  the  constants 
x^,  y\.  In  the  same  way  the  expression  v  takes  on  a  constant 
value  i\  when  x,  y  are  replaced  by  ^, ,  )\.  Thus  equation  (4) 
takes  on,  after  this  substitution,  the  form 

z/j  z^j  =  o (7) 

and  it  remains  to  be  proved  that  this  is  a  true  equation. 
By  hypothesis  [x^,y^)  lies  on  the  curve  (5).  Accordingly  its 
coordinates  satisfy  (5)  and  we  have 


and  from  this  the  truth  of  (7)  follows. 

We  have  thus  proved  that  (x^,  y^)  lies  on  (4),  and  by  pre- 
cisely the  same  method  we  show  that  ('^2jJ^'a)  does  so. 

In  order  finally  to  show  that  (^^3,  ^3)  does  not  lie  on  (4)  let 
us  indicate  by  ti^  the  value  of  the  expression  tt  when  the  vari- 
ables x,y  are  replaced  by  the  constants  x^,y^,  and  by  z^,  the 
value  of  V  after  the  same  substitution.  The  result  of  substi- 
tuting ^3,>',  in  (4)  is  then 

^^3  ^s  =  o (8) 

This  equation,  however,  is  not  true,  since,  the  point  (^3,^3) 
lying  on  neither  (5)  nor  (6),  neither  u^  nor  v^  is  zero. 

The  result  of  this  section  may  be  stated  by  saying  that  if  we 
have  two  curves  whose  equations  are  so  written  that  the  right- 
hand  -members  are  zero,*  the  two  curves  may  be  represented 
by  a  single  equation  obtained  by  multiplying  these  equations 
together. 

*  This  restriction  is  very  important,  for  if  the  right-hand  members  are  not  both  zero 
the  fact  here  stated  will  not  be  true.  Thus  if  we  have  the  two  lines  x  =  \,  x  =  2  and 
multiply  these  equations  together  as  they  stand,  the  resulting  equation  x"^  =  2  repre- 
sents the  two  wholly  different  lines  x  =  ±  V2.  In  order  to  get  an  equation  represent- 
ing the  two  lines  we  started  with  we  must  write  their  equations  x  — 1=0,  ^  —  2  =0, 
and  then  multiply;  getting,^^  —  3^  4-  2  =  o  as  the  equation  of  the  pair  of  lines. 


LOCI.  Ill 

EXAMPLES. 

What  are  the  loci  of  the  following  equations  ? — 

(i.)  x"^  —y  =  o.  (2.)  X'  +  2  xy  —3^  =  0. 

(3.)  x'^y  +y'  —  ^y  —  o.  (4.)  Jt^*  —  /  —  9  ^'  +  9/  =  o. 

(5.)  Prove  that  if  «,  v,  w  are  three  expressions  in  x^  y,  the 
equation   7^vu'  =  o    has  as  its  locus   the  three  curves  :  u  =  o, 

V  =  O,  IV  =  o. 

(6.)  Show  that  the  first  members  of  the  following  equations 
can  be  factored  : 

ax'^  +  bx  +  c=^o^ 
ax"^  +  b  xy  +  cy"^  =  0. 
What  do  these  equations  represent  ? 

60.  In  the   last  section  we  have  seen  what  the  result  is  of 

multiplying  together  the  two  equations 

u  =^  o •  •  (i) 

^'  =  o (2) 

Let  us  now  consider  what  the  result  of  adding  these  equa- 
tions is.  That  is,  we  wish  to  find  out  as  much  as  we  can  about 
the  locus  of  the  equation 

u  ■\-v^o (3) 

Let  (^i,jV,)  be  any  point  common  to  the  two  curves  (i)  and 
(2),  i.e.  any  point  of  intersection  or  of  contact  of  these  curves, 
and  let  us,  as  in  the  last  section,  denote  by  u^  and  i\  the  values 
of  the  expressions  u  and  v  when  the  variables  x^y  are  replaced 
by  the  constant  values  x^^y^.  Now  since  (:v,,  y^  lies  on  (i)  we 
have 

7/,  =  o. 


112  ANALYTIC    GEOMETRY. 

Since  (jc,,  jv,)  lies  on  (2)  we  have 

v^  =  o. 
Accordingly 

u,  -{-v^  =  o, 

i.e.  {x^,y^)  lies  on  (3).  Thus  we  see  that  (3)  passes  through 
all  the  points  common  to  (i)  and  (2). 

Now  let  (^,,  y,)  be  a  point  of  (i)  which  does  not  lie  on  (2). 
Then,  letting  «,,  v^  denote  the  values  of  u  and  Z'  when  the  vari- 
ables X, y  are  given  the  values  -x^^y^,  we  have 

u,  =  o, 

^,  7^  o- 
Therefore 

u^  +  v^  ^  o. 

That  is  (.r„  y^  does  not  lie  on  (3). 

In  the  same  way  we  see  that  a  point  (x^,  j',)  which  lies  on 
(2)  but  not  on  (i)  will  not  lie  on  (3). 

Thus  we  have  established  the  fact  that  if  we  add  together 
the  equations  of  two  curves,  the  resulting  equation  will  have  a 
locus  which  passes  through  all  the  points  common  to  the  two 
curves,  but  meets  neither  curve  in  any  other  point. 

We  can  easily  throw  this  result  into  a  slightly  more  general 
form.  The  equations  of  the  curves  (i)  and  (2)  may,  if  we 
wish,  be  written  in  the  form 

a  u  =^  o (4) 

bv-o •  .  (5) 

where  «,  b  are  any  constants  other  than  zero.  By  applying 
the  principle  just  established  to  the  curves  (4)  and  (5)  we  get 
the  theorem  : 


LOCI.  113 

Ifu=^o  and  v  =  o  are  the  equations  of  two  gii  en  curves^ 
then 

a  u  -^  bv=  o 

represents  a  curve  which  passes  through  all  the  points  common  to 
the  two  given  curves,  and  ?neets  neither  of  them  in  any  other 
point. 

Just  what  curve  of  this  sort  we  get  depends,  of  course,  on 
the  values  of  the  constants  a  and  b. 

61.  As  an  illustration  of  the  way  in  which  this  principle  can 
be  applied  let  us  consider  tlie  problem  of  finding  the  equation 
of  the  common  chord  of  two  intersecting  circles.  Consider,  for 
instance,  the  two  circles. 

x^  +  y^  —  2  X  —  4y  +  4  =  o (i) 

x'^+y^  —  4x  —  2y  +  i=o.  .  .   .  '.  .  .  .   (2) 

These  circles  intersect,  as  the  reader  will  see  if  he  draws  a 
rough  figure.  In  order  to  find  the  equation  of  their  common 
chord  let  us  denote  the  first  member  of  (i)  by  «,  the  first 
member  of  (2)  by  v.     Then  the  equation 

a  u  -h  b  V  =  o (3) 

will  represent  some  curve  through  the  two  points  of  intersec- 
don  of  (i)  and  (2).  But  it  is  clear  that,  in  general,  the  equa- 
tion (3)  is  of  the  second  degree  and  therefore  does  not  repre- 
sent a  straight  line.  If,  however,  we  give  to  a  the  value  i,  to 
b  the  value  —  i,  the  terms  of  the  second  degree  cancel  and 
the  equation  (3)  becomes 

u  —  V  =  o 
or 

2X  —  2y  -^  S  =  o (4) 


114  ANALYTIC    GEOMETRY. 

This,  being  of  the  first  degree,  represents  a  straight  line,  and, 
being  merely  a  special  case  of  (3),  this  line  must  pass  througii 
the  points  of  intersection  of  (i)  and  (2).  Thus  (4)  is  the  equa- 
tion we  were  seeking. 

The  equation  (4)  was  obtained  by  subtracting  the  equation 
of  one  of  the  given  circles  from  the  other  ;  and  it  is  clear  that 
the  same  reasoning  applies  to  any  similar  case.  That  is,  to 
find  the  equation  of  the  common  chord  of  two  given  intersect- 
ing circles,  write  the  equations  of  these  circles  so  that  the 
coefficients  of  x"^  and  j''^  are  t,  and  then  subtract  one  equation 
from  the  other. 

EXAMPLES. 

(i.)  A  circle  has  its  centre  at  the  origin  and  radius  i,  and  a 
second  circle  has  its  centre  at  the  point  (  —  1,2)  and  radius 
2.     Find  the  equation  of  the  common  chord  of  these  circles. 

(2.)   Show  that  the  three  lines 

2  X  +  y  ~  ^  =■-  o, 
X  -  2y  +  5  =  o, 
4x  —  2,y  +  T  =  0 

meet  in  a  point. 

[Suggestion:  Show  that  if  the  first  two  equations  are  de- 
noted by  «  —  o,  7'  =  o,  tlie  third  can  be  written  in  the  form 
^  //  +  <^  7'  =  o  ] 

(3.)  Find  the  equation  of  the  line  which  connects  tlie  origin 
with  the  point  of  intersection  of  the  lines 

X  —  2y  —  3  =  0, 

2  X  —  J'  +   5   =r  O. 

[Suggestion  :  Determine  the  constants  a  and  b  in  tlie  equa- 
tion a  u  -\-  b  Z!  ^=  o,  so  as  to  eliminate  the  constant  term.] 


LOCI.  115 

(4.)   Prove  that  the  three  common  chords  of  three  intersect- 
ing circles  meet  in  a  point. 

[Suggestion  :  Let  the  circles  be 

x^  +  y""  +  a^  X  +  if^y  +  <^i  —  o> 
^'  +  y  +  a^  X  +  t?^y  +  c^  =  o, 
x"^  +  f  +  a^  X  -\-  b^  y  +  c^  =  o. 

Find  their  common  chords  by  §  61,  and  then  use  the  method 
of  example  2.] 


CHAPTER  VII. 


THE   CONIC    SECTIONS. 


62.  An  ellipse  is  a  curve  generated  by  a  point  moving  in  a 
plane  so  that  t^^  sum  of  its  distances  from  two  fixed  points  in  that 
plane  is  constant. 

An  hyperbola  is  a  curve  generated  by  a  point  moving  in  a  plane 
so  that  the  difference  of  its  distances  frojn  two  fixed  points  in  that 
plane  is  constant. 

A  parabola  is  a  curve  generated  by  a  point  moving  in  a  plane 
so  that  at  each  instant  its  distances  from  a  fixed  point  and  a  fixed 
line  in  that  plane  are  equal. 

These  three  curves  are  called  the  conic  sections,  because,  if  a 
right  circular  cone  is  cut  by  a  plane  which  does  not  pass 
tlirough  the  vertex  of  the  cone,  the  section  will  be  bounded 
by  one  of  these  curves. 

We  will  find  the  equations  of  the  conic  sections. 

Y 

63.  In  finding  the  equation 
of  the  ellipse,  we  will  take  the 
straight  line  joining  the  two 
fixed  points,  which  are  called 
foci,  for  the  axis  of  x,  and 
the  point  midway  between 
the  foci,  called  the  centre,  for 
the  origin. 

In  the    figure,  F^  and   F 
are  the  foci ;  P  is  any  point 


THE    CONIC    SECTIONS.  II7 

of  the  curve;  and  r'  and  r,  called /^^^/  radii^  measure  the  dis- 
tances of  P  from  F'  and  F^  respectively. 

Let  2  c  represent  the  constant  distance  between  the  foci,  and 
2  a  the  sum  of  the  focal  radii,  which  is  constant  by  the  defini- 
tion of  the  curve.  In  the  triangle  F'  FP  the  side  F'  F  —  2c 
is  less  than  2  a,  the  sum  of  the  other  two  sides.     Tlierefore 

c  <  a     (1) 

The  coordinates  of  F'  and  F  are  {— c,  o)  and  {c,  o),  and 
the  equation  which  expresses  the  definition  of  the  locus  is 

r'  +  r  ■=  2  a. 
Expressing  /  and  r  in  terms  of  x,  y.  and  c,  by  §  10, 

r'  =  ^{x^  cY  +J-, (2) 

r  ^  y/yx  -  ^^f~Vy' (3) 

Substituting  these  values,  we  have  the  equation  of  the  ellipse, 
a/(^  +  cf  .f  /  +  ^/{x  -cy  ^  y  =  2a, (4) 

which  will  be  in  a  more  convenient  form  when  cleared  of  radi- 
cals. 

Equation  (4)  may  be  written 

V{x  +  cy^  +f  =  2  a-  V{x-cy  +f', 

squaring, 

.  {^x  +  cy  +  y''  =  4a^  -  4a  V{x  -  r)-  +  y^  +  (x  -  cf  +  / ; 
expanding,  combining,  and  transposing, 

4  a  ^/{x  —  c)~  -Y y'^  —  \a^  —  ^c X'y 
dividing  by  4,  and  squaring  again, 

d^  x^  —  2  or  c  X  -\-  a}  c^  4-  (r  y^  —  a^  —  2  a^  c  x  -\-  c^  x^  \ 

reducing, 

{a^  -  c')  x'  +  a'y'  =  a''  (^^  -  c') (5) 

We  will  represent  the  constant  quaniity  {a*  —  c^),  which,  as  we 
see  from  (i),  is  positive,  by  d"^;  with  this  notation,  (5)  becomes 


ii8 


ANALYTIC    GEOMETRY. 


or,  dividing  by  d^  b"^. 


b'^x'  +  a'^y'  =  a^b\ 


X'      y* 

~2    +    A2 


a"      b 

the  usual  form  of  the  equation 


(6) 
(7) 


64.  With  the  same  clioice  of  axes,  the  equation  of  the  hyper- 
bola may  be  found  in  a  similar  manner. 

In  this  case  we  represent  the  constant  difference  between  the 
focal  radii  by  2  a.  Since  the  side  F'  F  —  2  c  oi  the  triangle 
F'  FP  is  greater  than  the  difference  2  a  oi  the  other  two 
sides,  we  have 

0  a (i) 

Accordingly  we  use  F'  as  an  abbreviation  for  /  —  ^^ 

With  this  notation  show  lliat  llie  equation  of  the  hyperbola  is 


(2) 


Note.  Since  the  equations  of  the  ellipse  and  hyperbola  differ 
only  in  the  sign  of  z^"",  the  results  of  smiilar  operations  performed 
on  each  will  differ  in  the  same  way.  One  of  these  results  may, 
therefore,  be  obtained  from  the  other  by  changing  the  sign  of  ^^, 
a  method  which  will  enable  us  to  infer  many  properties  of  the 
hyperbola  from  the  corresponding  properties  of  the  ellipse,  or 
vice  versa. 

65.  In  finding  the  equation  of 
Xheparabo/a,  we  will  take  the  fixed 
straight  line,  called  the  directrix, 
as  the  axis  of  y,  and  the  perpen- 
dicular upon  it  from  the  fixed 
point,  or  focus,  as  the  axis  of  x. 

In  the  figure  O  V  is  the  direc- 
trix,   F  the    focus,    and    F    any 
^    point   of   the   curve.      Letting  m 


THE    CONIC    SECTIONS.  IIQ 

represent  the  constant  distance  of  the  iooM^from  the  directrix, 
the  coordinates  of  F  are  (w,  o). 

The  equation  which  expresses  the  definition  of  the  locus  is 

FP  ^  EP. ,  .  (i) 

By  §  lo 

FP  ^  ^/{x  -  my  +  /, 
Also 

FP  =  O  M  ^x. 

Substituting  these  values  in  (i),  we  have,  after  squaring, 

{x  —  77iy-  -i- y^  =  x'^f    .   .   .  c (2) 

which  reduces  to 

y^  —  2  m  X  -{-  m^  =  o, (3) 

the  equation  sought. 

66.  A  simpler  form  of  the  equation  of  the  parabola  may  be 
found  by  changing  the  origin  to  the  point  midway  between  the 
focus  and  directrix. 

The  formulas  for  transformation  are 

m 
x  =  x'-\.-,    y=y, 

and  the  new  equation  is,  therefore, 

which  becomes,  after  reducing  and  dropping  the  accents, 

f-  —  2  m  X, 
the  usual  form  of  the  equation  of  the  parabola. 

Note.  The  student  will  remember  that  we  have  already 
found  the  equations  of  the  conic  sections  in  the  chapter  on  loci. 
It  will  be  useful  for  him  to  compare  the  definitions  of  the  conies 
and  their  equations  with  examples  7,  8,  14,  of  §  55. 

6y.  We  will  now  investigate  the  forms  of  the  conic  sections 
by  discussing  their  equations. 


y''^  —  2  m  (  ^^  +  —  I  +  w^ 


t^O  ANALYTIC    GEOMETRY. 

Beginning  with  the  ellipse,  we  find  its  intercepts  on  O  X  to 
be  a  and  —  a\  on  6>  K,  b  and  —  b.  Solving  equation  (6),  §  63, 
forjv,  we  have 

or 

y=  ±^7^1^?, (I) 

from  which  it  appears  that  with  each  value  oi  ^  there  are  two 
values  of  _>',  equal  numerically  but  opposite  in  sign;  this  shows 
that  the  curve  is  symmetrical  with  respect  to  OX.  Similarly 
solving  for  x^  we  have 

X=±j  V'^^ (2) 

From  this  equation  we  infer  that  the  curve  is  symmetrical 
with  respect  to  O  Y. 

From  (^i)  we  see  that  the  values  of  y  are  real  when  x  is  less 
than  a,  null  when  x  equals  a,  and  imaginary  when  x  is  greater 
than  a,  in  numerical  value  ;  also,  as  x  increases,  y  must  decrease. 

From  these  considerations  we  infer  that  the  curve  cuts  the 
axis  of  y  at  two  points  equally  distant  from  the  origin ;  that  it 
extends  from  these  points  both  to  the  right  and  the  left,  gradu- 
ally approaching  the  axis  of  x  in  each  direction,  and  meeting 
that  axis  at  points  whose  distances  from  the  origin  are  a  and 
—  a;  and,  also,  that  there  are  no  points  of  the  curve  farther 
from  O  Kthan  the  points  in  which  it  cuts  OX, 

Similarly,  by  examining  equation  (2),  we  see  that  the  curve  is 
limited  in  the  direction  of  the  axis  of  y  by  two  points  at  dis- 
tances b  and  —  b  from  the  origin,  lying  wholly  between  these 
points. 

Since  the  ellipse  is  symmetrical  with  respect  to  each  axis,  it 
must  be  symmetrical  with  respect  to  their  intersection,  the  origin 
or  centre  of  the  curve  ;  therefore  every  chord  drawn  through 
the  centre  is  bisected  at  that  point.     It  is  easy  to  prove  this 


THE   CONIC   SECTIONS. 


I2t 


analytically ;  for  the  equation  of  any  line  through  the  origin  is, 
by  §  25, 

y  =  ^x>   (3) 

and  the  equation  of  the  ellipse  is 

,2 

(4) 


a-       A' 


Now  if  (^',  y)  is  one  point  of  intersection  of  the  loci  of  (3) 
and  (4),  it  is  evident,  from  the  form  of  these  equations,  that 
( _  x\  —  y^)  will  also  satisfy  both  and  be  the  other  point  of 
intersection.     Since  the  distance  of  each  of  these  points  from 


the  origin  is  V-^'"  + J''^  the  origin  or  centre  must  be  the  middle 
point  of  each  chord  drawn  through  it. 

B^  are  at  distances  a^  —  a, 
b,  —  b  from  O  respec- 
tively. The  ellipse  must 
cut  the  axes  at  these 
points,  and  will  be  in- 
scribed in  the  rectangle 
whose  sides  are  parallel 
to  the  axes  and  pass 
through  the  points  A,  A', 
B,  B'.  Its  form,  which  is  shown  in  the  figure,  may  be  deter- 
mined with  more  precision  by  plotting  a  number  of  points,  or  by 
the  following  method  based  on  the  definition  of  the  curve. 

Let  the  extremities  of  a  flexible  string,  whose  length  is  2a,  be 
fixed  at  the  foci,  F'  and  F;  let  F,  the  point  of  a  pencil,  move  in 
the  plane,  keeping  the  string  tight ;  its  path  must  be  the  ellipse 
sought. 

The  transverse,  or  major,  axis  of  an  ellipse  is  that  part  of  the 
straight  line  through  the  foci,  contained  by  the  curve ;  it  is  repre- 
sented in  the  figure  by  A^  A. 


122  ANALYTIC    GEOMETRY. 

The  co?ijugate,  or  minor,  axis  is  that  part  of  the  perpendicular 
to  the  major  axis,  through  the  centre,  which  is  included  by  the 
curve  ;  it  is  shown  in  the  figure  as  B^  B. 

The  student  has  probably  noticed  that  a,  the  half-sum  of  the 
focal  radii,  is  equal  to  the  semi-major  axis  of  the  ellipse,  and  that 
b  is  the  semi-minor  axis.  In  §  63  we  introduced  b  as  an  ab- 
breviation for  ^/ a"  —  ^';  hence  b  <  a.  Thus  we  see  the 
reason  for  the  terms  major  axis  and  minor  axis. 

These  results  may  be  obtained  directly  from  the  figure ;  for 
by  the  definition  of  the  curve, 

F^  A-\-  FA  z=2a, 

and,  therefore,  since  A'  F'  equals  FA, 

A'  A  =  2  a  and  O  A  =  a  ; 
also,  since 

F'  B  ^-  FB  =  2  a, 

and  i^'^  equals  FB,  we  have,  from  the  right  triangle  O  FB, 


OB  =  a/^2  -  c'  =  b. 

The  eccentricity  of  an  ellipse  is  the  ratio  oi  c  to  a ;  this  will 
always  be  less  than  unity  [v.  (i),  §  63]. 

EXAMPLES. 

(i.)    The  distance  between  the  foci  of  an  ellipse  is  6,  and  the 

sum  of  the  focal  radii  of  any  point  is  10 ;  find  its  equation. 

x"  1'^ 

Ans.   —  +  -^  =    I. 

25        16 

(2.)    Determine  the  semi-axes  of  the  ellipse 

x"  +  2/  =  8. 
This  may  be  put  in  the  form  of  equation  (7),  §  63,  by  dividing 
each  term  by  8,  the  constant  term.     This  gives 


THE   CONIC    SECTIONS. 


123 


which  is  the  equation  of  an  ellipse  for  which 
cr  —  8  and  <^^  =  4. 
The  required  semi-axes  are,  therefore, 

a  —  2  V^  and  b  =  2, 
(3.)    What  are  the  semi-axes  of  the  ellipse ' 

16  A-^  +  257^  =  1600? 
Letting  e  represent  the  eccentricity  of  the  elllpserSfeew-WM 


(4.)    Determine  a,  b,  and  e  for  the  ellipse 


169       25 


c^ 

rS 


(5.)    The  equation  of  a  chord  of  the  ellipse 
^2  _^  7  /  _  16  =  o, 


IS 


^-7j  +  4  =  o; 
find  the  length  of  the  chord. 


rr-3 


c^ 


Aas^Ji,/t^sz 


(6.)  The  /cz///i-  rectum,  or  parameter,  of  a  conic  is  the  breadth 
of  the  curve  through  a  focus ;  it  is,  therefore,  equal  to  twice  an 
ordinate  erected  at  a  focus. 

Show  that  the  latus  rectum  of  an  ellipse  is  equal  to  — . 

(7.)  Prove  that  the  minor  axis  of  an  ellipse  is  a  mean  propor- 
tional between  the  major  axis  and  latus  rectum. 

(8.)  Find  the  equations  of  the  focal  radii  drawn  to  a  point  P^ 
of  the  ellipse. 


Ans. 


\y,x-  {x,  +c)  y  +  cy^  =  0, 
\  y^x  -  {x^  -  c)  y-c y^z=  o. 

(9.)    Find  the  points  of  intersection  of  the  circle  and  ellipse 
whose  equations  are 


124  ANALYTIC    GEOMETRY. 

x^  -^  y^  —  lo,  x^  +  y  y^  =  i6c 

Ans.   (3,  i),  (3,-1),  (-3,  i),  (-3»  -i)- 
(10.)    Find  the  points  of  intersection  and  the  length  of  the 
common  chord  for  the  curves  whose  equations  are 

x^  +  y'^  -{■  20  X  -\-  ^o  =  o,     x'^  -{-  J  y'^  —  16  =  o. 

Ans.    (-3,  i),  (_  3,  _  i),  2. 

(11.)  Prove  that  the  semi-major  axis  of  an  ellipse  is  a  mean 
proportional  between  the  intercepts  on  that  axis  of  the  lines 
which  join  the  extremities  of  the  minor  axis  with  any  point  of  the 
curve. 

68.  It  is  worth  while  for  us  to  notice  that  the  circle  is 
merely  the  special  case  of  the  ellipse  in  which  the  two  foci 
coincide,  for  in  this  case  c  —  o^b  —  a^  and  the  equation  of  the 
ellipse  reduces  to  the  familiar  form  of  the  equation  of  a  circle. 

It  may  be  noticed  that  the  eccentricity  c  of  the  circle  is 
zero. 

69.  To  investigate  the  form  of  the  hyperbola,  we  first  find 
its  intercepts  on  the  axes.  Obtaining  these  from  the  equa- 
tion (2)  of  §  64,  we  find  the  intercepts  on  the  axis  of  x  to  be 
a  and  —a;  those  on  the  axis  of  j  to  be  /^  V—  i  and  —  b  V"  i, 
which  are  imaginary.  The  hyperbola,  therefore,  does  not  cut  the 
axis  of  _y,  but  since  it  has  points  on  each  side  of  that  axis,  it  must 
comprise  at  least  two  parts  or  bra7iches. 

Solving  equation  (2),  §  64,  for  y  and  x,  we  have 

y  ^±-    \/x-  -^a\ (i) 


x  =  ±-  Vb-'  +y\ (2) 

equations  which  show  that  the  curve  is  symmetrical  with  respect 
to  each  axis. 

From  equation  (i),  we  see  that  the  values  of  ^  are  imaginary 


THE    CONIC    SECTIONS. 


125 


when  X  is  less  than  a,  null  when  x  equals  a,  and  real  when  x  is 
greater  than  a,  in  numerical  value. 

These  facts  show  that  there  are  no  points  of  the  curve,  either 
above  or  below  O  X,  between  two  points  at  distances  a  and  —  a 
from  the  origin,  while  beyond  these  points  the  curve  extends 
indefinitely  in  each  direction  of  that  axis. 

We  learn  from  (2),  that  all  values  of  y  give  real  values  of  x  ; 
consequently  we  infer  that  the  curve  runs  without  interruption 
in  the  direction  of  the  axis  of  7,  extending  indefinitely  in  the 
positive  and  negative  directions. 

In  either  (i)  or  (2),  we  ob- 
serve that  X  and  J'  must  increase 
together ;  therefore,  the  curve 
recedes  from  both  axes  at  the 
same  time. 

We  may  show,  as  for  the 
ellipse,  that  every  chord  drawn 
through  the  centre  of  an  hyper- 
bola is  bisected  by  the  centre. 

The  curve  represented  in  the  figure  agrees  with  these  results. 
It  may  be  constructed  by  finding  a  number  of  its  points, 
or,  by  a  method  *  based  on  the  definition  of  the  curve,  as 
follows:  — 

A  pencil  F  is  tied  to  a  point  near  the  middle  of  a  piece 
of  string.  Two  pegs  are  driven  into  the  drawing-board  at  the 
points  7^  and  F\  and  the  string  is  looped  around  these  pegs 
in  the  manner  indicated  in  the  figure.  The  two  parts  of  the 
string  are  grasped  in  one  hand  at  H  and  drawn  away  from  F'. 
The  pencil  P  will  then  describe  an  arc  of  an  hyperbola  with 
foci  at  F  and  F\  for  the  two  distances  F  F  and  F'  F  are 
evidently  decreased  by  the  same  amounts,  so  that  their  differ- 

*  See  pnpers  by  Ransom  and  Huntington  in  the  Annals  of  Mathematics,  Second 
Series,  Vol.  Ill,  p.  164,  and  Vol.  IV,  p.  50. 


126  ANALYTIC    GEOMETRY. 

ence  remains  constant.     The  otlier  l^ranch  may  be  generated 
in    the    same    manner    by    placing    the    pencil   F   at    some 


point  of  the  other  branch  before  drawing  the  string  tight 
at  H, 

In  the  equation  of  the  hyperbola,  §  64,  we  used  b  to  represent 
^c^  —  a- ;  it  will  not,  as  for  the  ellipse,  be  the  length  of  the  in- 
tercepts on  O  V,  for  the  hyperbola  does  not  cut  O  V.  We  shall, 
however,  call  that  portion  of  the  axis  of  y  contained  between 
points  at  distances  b  and  —  b  from  the  origin^  or  centre,  the  con- 
jugate axis  of  the  hyperbola ;  its  relation  to  the  curve  will  be  seen 
hereafter. 

The  definitions  of  the  transverse  axis  and  eccentricity  are  the 
same  for  the  hyperbola  as  for  the  ellipse. 

The  transverse  axis  will  lie  between  the  two  branches  of  the 
curve  ;  the  eccentricity  must  be  greater  than  unity,  because  c  is 
greater  than  a  (v.  §  64),  and,  for  the  same  reason,  the  foci  F^ 
and  F  lie  one  within  each  branch  of  the  hyperbola. 


THE   CONIC   SECTIONS.  I27 

EXAMPLES. 

(i.)    The  distance  between  the  foci  of  an  hyperbola  is  10,  and 

the  difference  between  the  focal  radii  of  any  point  is  8  ;  find  its 

equation. 

Ans.  -^  -  —   =  1. 
16       9 

(2.)  Determine  the  semi-axes  and  eccentricity  of  the  hyper- 
bola 

9  x^  —  \6y'^  =  576. 

Using  the  same  method  as  in  Ex.  2,  §  67,  we  have 

a  =  8,  d  =  6,  e  =^. 
4 

(3.)    The  equation  of  a  chord  of  the  hyperbola 

^  x^  —  g  y'^  —  64  =  o 
is 

2  X    —  gy    —     8  =  0; 

find  the  length  of  the  chord,  and  determine  the  semi-axes  and 
eccentricity  of  the  hyperbola. 

A//S.    Length  of  chord  is  A/85  ;  e  -  — — . 

3 

2  //-^ 
(4.)    Show  that  the  latus  rectum  of  the  hyperbola  is  ;  and 

state  the  theorem  for  the  hyperbola  analogous  to  that  of  §  67, 
Ex.  7. 

(5.)  Show  that  the  equations  of  the  focal  radii  drawn  to  a 
point  J^^  of  the  hyperbola  are  the  same  as  the  equations  of  like 
lines  for  the  ellipse. 

(6.)    Find  the  points  in  which  the  following  curves  meet:  — 
x^      5  y^  x^      7  y'^ 

^"^"87  "'  ^"'sT  "  '• 

Ans.    (4,  ±3),  (-  4,  ±3)- 


128  ANALYTIC    GEOMETRY. 

70.  We  will  now  consider  the  equation  of  the  hyperbola  in 
polar  coordinates,  as  in  this  way  a  good  deal  of  light  will  be 
thrown  on  the  shape  of  the  curve. 

Using  the  formulae  of  §43,  we  have  as  the  equation  of  the 
hyperbola 

r'  cos*  0        r'  sin'  0  __ 


a'  d'         ""^' 


accordingly 


r=  ± 


=  ± 


cos" 


0       sin'  0 


\a^ 


cos  (p  ^^  ~2~  tan^  0 

Using  the  upper  sign  here  and  starting  with  the  value  0  =  o, 
we  have  r  =  a.  As  0  increases,  the  denominator  of  r  de- 
creases, and  therefore  r  itself  increases.  When  0  has  the 
value  00  for  which 

tan  00  =  - ' 

the  denominator  of  ;'  vanishes.  Accordingly  as  0  increases 
and  approaches  the  value  0^  as  a  limit,  r  increases  indefinitely. 
When  0  is  greater  than  0^  the  expression  under  the  radical 
sign  is  negative  and  r  is  therefore  imaginary.  Thus  we  see 
that  in  the  first  quadrant  the  hyperbola  lies  wholly  below  the 

line  through  the  origin  with  slope  -•     In  a  similar  manner  we 

could  discuss  the  shape  of  the  curve  in  each  of  the  other 
quadrants,  or  we  can  even  more  easily  deduce  the  result  from 


THE    CONIC    SECTIONS. 


129 


the  symmetry  of  the  curve.     By  either  method  we  find  that 
the   hyperbola  lies  wholly  between  the  two  lines  through  the 

origin  with  slopes  ±  — ,  while  any  line  through  the  origin  cuts 

the  curve  if  it  makes  with  the  axis  of  X  a  smaller  angle  than 
these  lines  do.     These  two  lines 

y  =  ±  —x 
a 

are  called  the  asymptotes  of  the  hyperbola. 

71.  The  most  remarkable  property  of  the  asymptotes  is  that 
the  hyperbola  approaches  them  indefinitely  as  it  is  extended  from 
the  centre,  but  never  reaches  them.  We  may  prove  this  prop- 
erty as  follows  :  — 

Draw  corresponding  ordinates    for    the   hyperbola    and    the 

asymptote  O  L.  These  are 
represented  in  the  figure 
by  M^  P^  =  y^  and  M^  P; 
—  y^.  The  equation  of 
O  Z,  the  asymptote  in 
the  first  and  third  quad- 
rants, is 

b 

y=-x, 
a 


From  this  equation  and  that  of  the  hyperbola,  we  obtaiff 


2       *'   /    2 


therefore 


j,^=^«^  =  *'; 


130  ANALYTIC    GEOMETRY. 

from  which  we  may  write 
or 


p.p:=y. 


' '    y;  +  )■, 

Now  as  P^  moves  away  from  the  centre,  jc,  increases  indefi- 
nitely, and  the  equations  of  the  hyperbola  and  asymptote  show 
that  the  accompanying  values  of  y^  and  jj^/  must  also  increase 
indefinitely.  The  value  of  P^P^'  must  therefore  approach 
zero  as  its  limit,  for  the  numerator,  ^^  is  constant,  while  the 
denominator  increases  indefinitely.  The  distance  of  P^  from 
the  asymptote,  measured  on  a  perpendicular,  is  equal  to  P^  P ^ 
multiplied  by  a  constant  quantity,  the  sine  oi P^P^' 0\  therefore 
this  distance  varies  with  P^P^',  and  approaches  zero  together 
with  it.     This  proves  the  theorem. 

It  may  be  shewn  in  like  manner  that  a  point  below  the  axis  of 
X,  or  one  on  the  other  branch  of  the  hyperbola,  approaches  one 
of  the  asymptotes  as  it  recedes  from  the  centre,  but  can  never 
really  reach  it. 

EXAMPLES. 

(i.)    Find  the  equations  of  the  asymptotes  of  the  hyperbola 

x^  -4/ =  36. 

Ans.   X  —  2  y  —  o,  x  -\-  2  y  —  o. 

(2.)  Show  that  the  distances  of  either  focus  from  the  asymp- 
totes equal  half  the  conjugate  axis. 

(3.)  Discuss  the  shape  of  the  ellipse  by  means  of  polar  co- 
ordinates. 

(4.)  Prove  that  if  from  any  point  on  an  hyperbola  a  line  is 
drawn  parallel  to  either  axis,  the  product  of  the  parts  cut  off 
between  the  point  and  the  asymptotes  equals  the  square  of  half 
that  axis. 


THE    CONIC    SECTIONS. 


131 


72.  Since  in  the  hyperbola  c  is  greater  than  a^  and  ^'  =  c^—a^, 
b  may  be  less  than  a^  equal  to  «,  or  greater  than  a^  according  as 

c 

a  is  greater  than,  equal  to,  or  less  than      /=, 

V2 
When  we  make  b  equal  to  a^  the  equation  of  the  hyperbola 

becomes 

oc"  -  /  =  a\ 

and  the  curve  is  called  an  equilateral  hyperbola. 

The  asymptotes  of  the  equilateral  hyperbola  are 

y  -  ±  X, 


and    since  these   lines   are  perpendicular   to  each  other,  this 
hyperbola  is  sometimes  called  the  rectangular  hyperbola. 

73.  To  investigate  tlie  form  of  the  parabola,  we  will  discuss 
the  equation  found  in  §  66. 

The  intercepts  of  this  curve  are  all  o.  If  we  solve  the  equa- 
tion for  y  and  x^  we  have 

(•) 


and 


y  —  -^  Vs  m  x^ 


X  = 


2  711 


(2) 


Y 

P 
E —p^ 

D  "Oi   F  X 


We  learn  from  equation  (i)  that  the 
curve  is  symmetrical  with  respect  to 
OX.  Also,  if  in  is  positive,  all  positive 
values  of  x  give  real  values  of  j-,  but 
negative  values  of  x  make  y  imaginary  ; 
in  this  case  the  curve  must  lie  wholly 
on  the  right  of  O  Y.  In  like  manner 
we  may  show  that  when  in  is  negative, 
the  curve  is  wholly  to  the  left  of  O  Y. 

From  (2)  we  see  that  all  values  of  j^ 


132 


ANALYTIC    GEOMETRY. 


give  real  values  of  x ;  consequently  the  curve  is  continuous  and 
indefinite  in  extent,  in  the  direction  of  the  axis  of  j. 

Either  (i)  or  (2)  shows  that  x  and  j  must  increase  together; 
therefore  the  curve  recedes  from  both  axes  at  the  same  time. 

The  parabola  represented  in  the  figure  agrees  with  these 
results.  It  may  be  constructed  by  points,  or  by  the  following 
method  based  upon  the  definition  of  the  curve :  — 

Let  the  leg  E  F^  of  the  right-angled 
triangular  ruler  E  F  G  slide  along  the 
directrix  D'  D ;  fix  one  end  of  a  string, 
equal  to  E  G,  at  the  focus  F,  and  attach 
the  other  end  to  the  ruler  at  G.  A 
pencil  point  F,  which  presses  the  string 
tightly  against  E  G,  must  move  on  a 
parabola. 

The  straight  line  through  the  focus  and 
perpendicular  to  the  directrix  is  called 
the  axis  of  the  parabola,  and  the  point 
in  which  it  cuts  the  curve  is  called  the 
ve7-tex  of  the  parabola.  The  equation  of  the  parabola  found  in 
§  66,  is  called  the  equation  referred  to  the  vertex  and  axis,  or 
simply,  to  the  vertex. 

EXAMPLES. 

(i.)    What  is  the  equation,  referred  to  the  vertex,  of  the  para- 
bola whose  focus  is  at  a  distance  7  from  its  directrix  ?     What  is 
the  equation  of  the  same  curve  referred  to  its  axis  and  directrix  ? 
Ans.  y^  —  \\x;    y"-  —  14:^  +  49  =  o. 

(2.)  Show  that  the  second  locus  constructed  in  §  17  is  a 
parabola  ;  how  far  is  its  focus  from  the  directrix  }  how  far  from 
the  vertex  ? 

(3.)    Find  the  latus  rectum  of  the  parabola. 

Ans.    2  m. 


THE    CONIC    SECTIONS. 


^33 


(4.)  Find  the  equations  of  the  chords  which  join  the  vertex 
to  the  extremities  of  the  latus  rectum,  the  vertex  being  the 
origin. 

Ans.  y  =  2  X,    y  -\-  2  x  =  o. 

(5.)  What  is  the  equation  of  the  circle  which  passes  through 
the  three  points  mentioned  in  the  last  example  ? 

Ans.    2  x^  -j-  2  y^  =  c^m  x. 
(6.)    The  equations  of  two  curves  are 

2,x'^  +  4^  =  48  and  ()x  —  2y^  —  o. 
What  are  the  curves,  and  where  do  they  intersect  ? 

74*  We  will  next  find  the  equations  of  a  tangent  and  a  normal 
^o  an  ellipse  at  a  point  /\  of  the  curve. 

y  In  the  figure,  P^  T 

is  the  tangent,  and 
P^  JV  the  normal. 
The  coordinates  of  P^ 
are  x^  =  O  M^  and 
y,^M,P,. 

Using  the  same  me- 
thod as  in  §  49,  we 
have,  corresponding  to 
equation  (4)  of  that  section,  the  following  equations,  obtained 
from  equation  (6),  §  63: — 

b'x;"  +  a'y^  ^a-'b\ (i) 

b\x^  +  hY  +  a\y,  +  ky=aH' (2) 

Subtracting  (i)  from  (2),  we  may  write 

21?' hx^  +  b'  h''  +  2a^ky^  +  k'a'  =  o, 
from  which  we  obtain 


134  ANALYTIC    GEOMETRY. 

Taking  the  limit  of  this  as  ^  and  /i  both  approach  zero,  we 
have 

P  X 


and 


I         a'^y^ 


^^-y,  =^ (5) 

The  equation  of  the  tangent  is,  therefore, 

/y'x,  , 

y-yx  ^-^/-^-^i)'  •  • (6) 

and  of  the  normal 

^  -  ^1  =  1^  (^  -  ^i) (7) 

Clearing  (6)  of  fractions,  and  transposing,  we  have 
b^  x^x  ■\-  a'^j\  y  —  b^  x^  +  ^^  J'l^ 
which,  by  equation  (i),  may  be  written 

b'^  x^x  +  a^y^y  —  d^b'^, (8) 

or 

-^+7^='' (9) 

the  usual  form  of  the  equation  of  the  tangent. 

Clearing  equation   (7)  of  fractions,   we  may  write  it  as  fol- 
lows :  — 

d^ y^  X  -  Ir  x^y  -  {a^  -  b'^)  x^y^  =  o, (10) 

a  form  which  may  readily  be  obtained  from  (8)  by  the  method  of 
§  34- 

EXAMPLES. 
(i.)    Find  the  equations  of  tangent  and  normal  at  (i,  2)  to  the 
ellipse 

3  x''  -f  4/  =  19. 


THE    CONIC    SECTIONS.  135 

Since  the  coordinates  of  the  point  satisfy  the  equation  of  the 
curve,  the  point  lies  on  the  ellipse,  and,  finding  the  semi-axes, 
we  may  write  the  equations  of  tangent  and  normal  as  follows :  — 

X         2  y 

1^  +  -yT   =  '• 
or 

3^  +  87  =  19, 
and 

8^  -  3  J  =  2. 


x^       y^ 


(2.)    Find  the  equations  of  tangents  and  normals  to  the  ellipse 

12+    6   -    ' 

at  all  points  of  the  curve  whose  abscissas  are  numerically  equal 
to  2. 

(3.)  Calculate  the  intercepts  on  the  axis  of  x  of  the  tangent 
and  normal  drawn  at  a  point  J^^  of  the  ellipse. 

Ajis.  —  ,    — — 5—  X,  or  e^  x.. 

x^  a^        ^  ^ 

(4.)    Show  that  the  lengths  of  the  sub-tangent  and  sub-normal 

(i~  —  X  '^  b^ 

for  a  point  F^  of  the  ellipse  are  ^    and   -^^j,  respectively. 

(v.  §  49,  Ex.  12,  and  the  figure  of  this  section,  where  J/,  T  and 
N M^  are  the  required  lines). 

(5.)  By  means  of  these  results,  prove  that  if  a  series  of  el- 
lipses have  the  same  major  axis,  tangents  drawn  to  them  at  cor- 
responding points  (i.  e.,  points  which  have  the  same  abscissa) 
meet  on  the  axis  of  x. 

(6.)  Find  those  points  on  an  ellipse  where  the  tangent  is 
equally  inclined  to  the  axes. 


/  a"-  .  b^      \ 


136  ANALYTIC    GEOMETRY. 

(7.)  Find  the  equations  of  the  tangents  to  the  ellipse, 

which  are  incHned  at  an  angle  of  45^  to  the  major  axis. 

Ans,    x—y  +  2=0,  x—y  —  2  =  0. 
(8.)  Solve  the  last  example  when  45°  is  replaced  by  f  tt. 

Ans.    X  -\-  y  —  2  =0,  x-\-y  +  2=o, 
(9.)  Find  the  equation  of  a  tangent  to  the  ellipse 

-^  +  i:^  =  I. 

at  the  extremity  of  the  latus  rectum. 

Ans.   ±  Vs  X  ±  ^  y  =  g  a  represents  the  four  tangents. 
(10.)  From  the  points  where  the  circle  on  the  major  axis  is 
intersected  by  the   minor  axis    produced,   tangents   are   drawn 
to  the  ellipse ;  find  the  points  of  contact. 

Ans.    The  extremities  of  the  latera  recta. 
(11.)  Find  the  condition  which  must  be  fulfilled  that  the  line 


f  +Z  =  i 


+ 
m       n 


may  be  tangent  to  the  ellipse 


oc^      y 


~  +  I2  =  '• 


Ans.   __+__  =  I. 
m-      n^ 


(12.)  Prove  that  if  with  the  coordinates  of  any  point  of  an 
ellipse  as  semi-axes,  a  concentric  ellipse  be  described  with  its 
axes  in  the  same  direction,  the  lines  which  join  the  extremities 
of  the  axes  of  the  first  ellipse  are  tangent  to  the  second. 

75-  The  method  of  finding  the  equations  of  a  tangent  and 
normal  to  an  hyperbola  at  a  point  of  the  curve  \?  entirely  analo- 


THE    CONIC    SECTIONS. 


137 


or 


gous  to  that   shown  in  the 
last  section  for  the  ellipse. 
Show    by    this    method 
that    the    equation    of    the 
y^    tangent  at  J^^  is 


•^1  ^  _j',y_  J . 

a'         "If-      ''   ' 
and  of  the  normal. 


(j-7i)=-^j(^-^i), 


(0 

(2) 
(3) 


Note.  These  equations  evidently  differ  from  the  correspond- 
ing equations  for  the  ellipse  only  in  the  sign  of  b^,  according 
to  the  principle  stated  in  the  note  to  §  64. 


EXAMPLES. 

(i.)    Find  the  equations  of  tangent  and  normal  to  the  hyper- 
bola 


at  the  point  (3,  a/z). 


2  x^  —  3  j^  =  12, 


Ans.   \      ^^-A^/  =  4, 

(  V2  -^   +   27   =   5    V2. 


(2.)    Find  the  equations  of  tangents  and  normals  to  the  hyper- 
bola 


^2         y^ 


=  I. 


9       3 

at  all  points  of  the  curve  whose  ordinates  are  numerically  equal 
to  3. 

(3.)^  Show  that  the  expressions  for  the  intercepts  on  O  X  oi  a 
tangent  and  normal  at  a  point  J^^  of  the  hyperbola  are  the  same 
as  for  the  analogous  lines  in  the  ellipse. 


\ 


138  ANALYTIC    GEOMETRY. 

(4.)  What  is  the  property  of  hyperbolas  analogous  to  that 
stated  in  §  74,  Ex.  5  ? 

(5.)  Show  that  the  lengths  of  the  sub-tangent  and  sub-normal 
of  the  hyperbola  have  the  same  expressions  as  for  the  ellipse. 
because  the  change  in  the  sign  of  b^  does  not  affect  the  numeri- 
cal value  of  the  sub-normal. 

(6.)  Find  those  points  on  the  hyperbola  at  which  a  tangent  has 
an  inclination  of  45°  ;  of  135°. 

(7.)  Is  the  problem  of  the  last  example  possible  for  every 
hyperbola  ?     If  not,  what  are  the  exceptions  ? 

(8.)    Find  the  equations  of  normals  to  the  hyperbola 

gx^  -  2sy"  =  225, 

which  have  an  inclination  of  45°. 

(9.)    Find  the  equations  of  tangents  and  normals  to  an  hyper- 
bola at  the  extremities  of  the  parameters. 
Ans.    For  the  upper  extremity  of  the  right-hand  latus  rectum, 

ex  —  y  =  a,    ax  +  aey  —  e{d^-{-  b'^). 

(10.)    Find  the  condition  which  must  be  fulfilled  that  the  line 

X       y 

— -f  —  =  I 
m       n 

may  touch  the  hyperbola 

x^       y^ 

d'  ~  i^  ^  ^' 

76.  Let  us  find  the  equations  of  a  tangent  and  a  normal  to  a 
parabola  at  a  point  F^  of  the  curve.  Using  the  simplest  form  of 
the  equation  of  a  parabola 


THE    CONIC    SECTIONS. 


139 


which  gives 
and 


y"^  =  2  m  X  .  .  .  (i) 
we  have  for  the  equations 
from  which  to  find  the  value 
of  A., 

y{'  =  2mx^.  .  .  (2) 
and 

(7,+^)-^-2^K+^).    (3) 

PVom  these  we  easily  find 


2Jr  -\-  k 


X.-"^ 
^-J. 


m 


The  required  equations  are,  therefore, 


m 


or 
and 

or 


y-y^=-^(x  -x^\ 


y^x  +  my  =  {m  -\-  x,)  y\, 


.  (4) 

.(5) 
.  (6) 

.  (7) 

,  (8) 

.  (9) 
(10) 


EXAMPLES. 


(i.)    Find  the  equations  of  tangent  and  normal  to  the  para- 
bola 


at  the  point  (i,  2). 


r  =  4-^, 


Ans. 


^      (x  -y  +  I  =0, 

\x  +  y  -  2  =:  o. 


146  ANALYTIC    GEOMETRY. 

(2.)  Obtain  the  equations  of  tangents  and  normals  to  the 
parabola 

y  =.  8  ;^ 

at  all  points  of  the  curve  whose  abscissas  are  equal  to  8. 

(3.)  Show  that  the  intercepts  on  O  X  oi  the  tangent  and  nor- 
mal at  a  point  /\  of  the  parabola 

y'^  =  2  mx 

are  —  x^  and  m  +  x^,  respectively. 

(4.)  Find  the  lengths  of  the  sub-tangent  and  sub-normal  for 
the  parabola ;  show  that  the  first  is  bisected  at  the  vertex,  and 
that  the  second  is  constant. 

(5.)  Prove  that  the  tangents  at  the  extremities  of  the  latus 
rectum  are  equally  inclined  to  the  axes  of  x  and  y,  and  meet  the 
directrix  on  the  axis  of  ^  (v.  §  67,  Ex.  6). 

(6.)  Form  the  equations  of  normals  to  the  parabola  at  the 
extremities  of  the  latus  rectum  ;  show  that  they  are  mutually 
perpendicular  and  intersect  on  the  axis  of  the  curve.  What  kind 
of  quarilateral  is  bounded  by  these  normals  and  the  tangents 
mentioned  in  the  last  example  ?     What  is  its  area  ? 

r  2  {x  +y)  =2>^, 

Ans.    •<  2  {x  —  y)  =z  ^  m, 

(  Area  =  2  fn^. 

(7.)  Show  that  if  the  focus  of  a  parabola  is  the  origin  and  the 
axis  of  the  curve,  the  axis  of  x,  the  equation  of  the  tangent  at 
J^j,  a  point  of  the  curve,  is 

y^^y  z=  m  (x  -}-  x^  +  m). 


(8.)    With  the  same  axes  as  in  the  last  example,  find  the  tan- 
gents and  normals  at  the  extremities  of  the  latus  rectum. 

:F>'+W  =  o, 
±y  —  m  =  o. 


Ans.  < 
Ix 


THE    CONIC    SECTIONS. 


I4t 


(o.)  Write  the  equations  of  the  directrix  and  iatus  rectum,  re- 
ferred to  the  vertex. 

(lo.)  Show  that  the  tangent  at  any  point  of  the  parabola 
meets  the  directrix  and  Iatus  rectum  produced  at  points  equally 


distant  from  the  focus,  the  common  distance  being 
(ii.)  Find  the  condition  that  the  line 


may  touch  the  parabola 


a      b 

y^  —  2  m  X, 


Ans.    2  b^  +  a  771 
77.   Instead  of  finding  the  tangent  to  the  ellipse 


*>-■ 


(0 


at  the  give7i  poi7tt  (:r,,  j,),  it  is  sometimes  necessary  to  find  the 
tangent  to  this  ellipse  which  has  a  givoi  slope  A.  It  is  obvious, 
when  we  consider  the  shape  of  the  ellipse,  tiiat  there  are  two 
such  tangents.     Their  equations  can  be  found  as  follows. 

Denoting  the  coordinates  of  the   point  of  contact  of  one  of 
these  tangents  by  (^,,  J^.),  its  equation  may  be  written 


x^  X 


=  I, 


(2) 


P'rom  (3)  we  have 


X,  b'          h' 
y  = ^x  -\ 


y^a 


and  the  equation  of  the  tangent  can  be  written 


y  =  Xx  -\- 


(3) 

(4) 

" rrW 

CIVIL  ENGINE 

U.  ol  C. 

ASSl!fUTl(i\'  I 


ER1N( 


Mkl 


142  ANALYTIC    GEOMKTRV. 

It  remains  merely  to  replace  jVj  in  this  equation  by  its  value 
in  terms  of  A.  This  value  we  find  as  follows.  Squaring  (4) 
and  clearing  of  fractions,  we  have  I 

b'x^  -  a'X'y,'  =  o (6) 

On  the  other  hand,  since  (Xi,J\)  is  a  point  of  the  ellipse  (i), 
it  satisfies  this  equation,  and  we  have 

d'x;  +  a\v^'  =  a'd' (7) 

By   multiplying  (7)  by  ^'  and  subtracting  (6)  from  it  we  elimi- 
nate x^  and  get 

or 

Substituting  this  value  of  jk,  in  (5),  we  get  as  the  desired  equa- 
tion of  the  tangent 

y  =  Xjc  ±   Va'  X'  +  b' (8) 

The  two  signs  here  give  us  the  two  tangents  with  slope  A. 

In  the  same  way  we  find  that  the  equations  of  the  tangents 
to  the  hyperbola  which  have  the  slope  A  are 


y^Xjc  ±    Va'A:'  -  b' (9) 

It  should  be  noticed  that  this  equation  becomes  imaginary 

when  A  is  numerically  less  tliAt  -  ,  so  that  the   hyperbola  has 

a 

no  tangents  which  make  a  smaller  angle  with   the  transverse 
axis  than  that  made  by  the  asymptotes. 

In  the  case  of  the  parabola  it  is  easily  seen  that  there  is  only 
one  tangent  with  a  given  slope  A.  By  using  the  method  ex- 
plained above,  its  equation  is  found  to  be 

y  =  Xx+~ (10) 

2/L 


THE    CONIC    SECTIONS.  I43 

EXAMPLES. 

(i.)  Deduce  the  results  of  this  section  by  starting  from  the 
equation  y  ^=  X  x  -]-  p  and  imposing  the  condition  that  the 
points  in  which  this  line  cuts  the  conic  should  coincide. 

(2.)  A  right  angle  moves  so  that  both  of  its  sides  touch  an 
ellipse.     Find  the  locus  of  its  vertex. 

(3.)  Solve  the  last  problem  if  instead  of  an  ellipse  we  have 
a  parabola. 

(4.)  Find  the  equations  of  the  tangents  from  the  point 
(14,  i)  to  the  ellipse  x"^  +  4}''^  =  100. 

Suggestion:  Call  the  coordinates  of  the  point  of  contact  of 
one  of  these  tangents  {x^,y^),  and  then  determine  the  values 
of  these  coordinates. 

78.   Let  us  find  the  lengths  of  the  focal  radii  (r  and  r'  in  the 

figure  of  §  6;^)  drawn  to  a  point  /^j  of  the  ellipse. 

By  §  10,  we  have 

r^  =  (x,-cy+y,\ (I) 

r'-'  =  {x,  +cy+y,' (2) 

Also  we  know  that  the  coordinates  of  I'j^  must  satisfy  the  equa- 
tion of  the  ellipse,  and  therefore 

y,'  =  ^,(a^-x,% (3) 

Substituting  this  value  of  j'/^  '^^  equation  (i),  we  have  after 
expanding  and  collecting, 

r^  =  ""'-/  x,'-2cx,  +  c^^  +  P (4) 

c 
Now  a-  -  b'  -  r  and  therefore  c^  +  /^'^  =  a^ ;  also  e  =  -    and 

consequently  c  -  a  e.     We  may  therefore  write  equation  (4)  as 

follows  :  — 

r^  —  e^  x^  —  2  a  e  x^-^  a^ ', 
and 

r  —e  X.    —  a,  or  a  —  e  X, (5) 


144  ANALYTIC    GEOMETRY. 

As  we  only  wish  the  length  of  r,  we  shall  use  the  second  of 
these  values,  because  the  first  is  necessarily  negative,  e  being  less 
than  unity  and  x^  less  than  a. 

In  like  manner  show  that 

r^  —  a  ^  e  Xy (6) 

This  may  also  be  proved  from  (5)  and  the  definition  of  the 
curve. 

EXAMPLES. 

(i.)  Prove  that  if  on  any  ellipse  three  points  be  taken  whose 
abscissas  are  in  arithmetical  progression,  the  corresponding  focal 
radii  of  these  points  are  also  in  arithmetical  progression. 

(2.)  Prove  that  if  the  ordinate  of  any  point  of  an  ellipse  be 
extended  to  meet  the  tangent  drawn  at  the  extremity  of  the  latus 
rectum,  the  length  of  the  extended  ordinate  equals  that  of  the 
corresponding  focal  radius  of  the  point. 

(3.)  The  ordinate  of  any  point  of  an  ellipse  is  extended  to 
meet  the  circumscribed  circle,  and  a  tangent  is  drawn  to  the  cir- 
cle at  its  extremity  ;  prove  that  either  focus  is  equally  distant 
from  the  point  and  the  tangent. 

79.  We  may  easily  find  the  segments,  F^  TVand  N F^  cut  by 
the  normal  from  the  base  of  the  triangle  F^  P^  F,  in  the  figure 
of  §  74,  by  adding  and  subtracting  O  TV^and  c.     They  are 

F^  N  ^  a  e  -^e^  x„  JV  F  =  a  e  -  e^  x^. 

Comparing  these  values,  we  have 

F'  JV  _  a  e  +  e^  x^  _  a  a-  e  x^  _r* 
JV  F       a  e  —  e^  x^       a  —  e  x^      r 

Since  the  base  is  divided  into  segments  proportional  to  the 
adjacent  sides,  we  know,  from  Geometry,  that  tJie  angle  between 
the  focal  radii  drawn  to  any  point  is  bisected  by  the  normal  at  that 
point. 


THE    CONIC    SECTIONS.  I45 

The  normal  is  the  internal  bisector  of  the  angle  between  the 
focal  radii,  consequently  the  tangent  perpendicular  to  the  normal 
must  bisect  the  supplementary  external  angle. 

80.  For  the  hyperbola  we  may  show,  as  in  §  78,  that  for  the 
right-hand  branch  (when  x^^  is  positive)  the  focal  radii  are 

r'  =  e Xj^  +  a,  and  r  —  e  x^  —  a  ; (i) 

for  the  left-hand  branch  (where  x^  is  negative) 

r^  ^  —  e  x^—  a,  and  r  —  a  —  e  x^ (2) 

EXAMPLE. 

Show  that  the  theorems  stated  in  the  examples  i  and  2  of 
§78  are  true  for  the  hyperbola. 

81.  The  tangent  to  an  hyperbola  at  a  point  P^  will  be  found 
to  cut  the  base  of  the  triangle  F^  P^  P  {%  75?  Figure)  into 
segments 

a"  a"- 

pf  T^ae  +  -.    TP=ae  -—- 

Comparing  these  segments,  we  have 

P'  T  _a  e  x^  +  d^  _e  x^  j^  a  _r* 
T  P       a  e  x^  —  (T-      e  x^  —  a      r  ' 

The  tangent  therefore  bisects  internally  the  angle  between  the 
focal  radii  of  the  hyperbola  drawn  to  the  point  of  tangency,  and 
consequently  the  normal  at  that  point  bisects  the  supplementary 
external  angle. 

82.  The  results  of  §§  79  and  81  show  us  that  when  an  ellipse 
and  hyperbola  have  the  same  foci,  the  curves  cut  each  other  at 
right  angles.  For,  at  the  points  of  intersection,  the  curves  have 
the  same  focal  radii,  the  tangent  to  the  hyperbola  bisects  the  in- 
ternal angle  between  these  radii,  and  the  tangent  to  the  ellipse 
bisects  the  external  angle.     These  tangents  are  therefore  mutu- 


46 


ANALYTIC    GEOMETRY. 


ally  perpendicular;  and,  since  a  tangent  to  a  curve  at  any  point 
has  the  direction  of  the  curve  at  that  point,  we  learn  that  co7i- 
focal  co?iics  intersect  at  right  a?igles. 


83.  It  follows  from  the  definition  of  the   parabola,  that  the 
focal  radius  F F^  (v.  figure  of  g  76)  drawn  to  a  point  F^  of  the 
7n 


curve  is  equal  to 


Now  we  may  easily  show,   by  the 


results  of  Ex.  3,  §  76,  that   TF  is  equal  to  x^  + 


therefore. 


the  triangle  T  F F^  is  isosceles,  and  the  tange?it  to  a  parabola  at 
any  point  must  make  equal  angles  with  the  axis  and  the  focal  radius 
drawn  to  that  point. 

From  this  it  follows  that  if  at  any  point  of  the  curve  the  focal 
radius  is  drawn  and  a  line  parallel  to  the  axis,  the  tangent  and 
normal  at  that  point  bisect  the  external  and  internal  angles  be- 
tween these  lines. 

84.   If  we  describe  a  circle  on  the  major  axis  of  an  ellipse  as 
a  diameter,  corresponding  or- 
dinates  of  the  ellipse  and  cir- 
cle will  have  a  constant  ratio, 
b  :  a. 

In  the  figure,  y^  {=  ^i  F^) 
and  _>'/  (=  Ml  jP/)  are  corre- 
sponding ordinates,  because 
they  are  drawn  with  the  same 
abscissa  x^  (=  O  M^).  From 
the  equations  of  the  ellipse 
and  circle,  we  have  for  the 
values  of  these  ordinates 

y,  =  ^-^/^::l^*, 


I  _ 


v^ 


.  3  . 

'1     S 


THE    CONIC    SECTIONS.  147 

therefore,  remembering  that  a  and  x^  are  by  hypothesis  the  same 
in  these  equations, 

which  was  to  be  proved. 

It  may  be  shown  in  like  manner  that  when  a  circle  is  inscribed 
in  an  ellipse,  with  the  minor  axis  as  a  diameter,  corresponding 
abscissas  of  ellipse  and  circle  have  the  constant  ratio,  a  :  b. 

By  means  of  these  properties  v/e  can  construct  an  ellipse  by 
shortening  each  ordinate  of  the  circumscribed  circle  in  the  ratio 
^  :  ^,  or  by  lengthening  each  abscissa  of  the  inscribed  circle  in 
the  ratio  a  \  b. 

We  can  also  draw  a  tangent  at  any  point  of  an  ellipse  by 
drawing  the  tangent  to  the  circumscribed  circle  at  the  corre- 
sponding point,  and  joining  the  point  in  which  this  tangent  meets 
the  major  axis  produced,  with  the  point  of  the  ellipse  at 
which  the  tangent  is  to  be  drawn  ;  for  the  circumscribed  circle 
may  be  regarded  as  one  of  the  series  of  ellipses  mentioned  in 
§  74,  Ex.  5. 

EXAMPLE. 

Two  tangents  to  the  circle  whose  equation  is 

x^  -f  y-  —  100 

meet  in  the  point  (14,  2)  ;  find  the  equations  of  tangents  to  the 
corresponding  points  of  the  ellipse 

or  +  4  y^  =  1 00. 

Ans.    3  .r  +  8  J'  =  50,   2  ^  -  3  j  =  25. 

85.  By  means  of  the  theorem  of  §  84,  we  may  calculate  the 
area  of  an  ellipse. 

Let  A' A^  in  the  figure  of  §  84,  be  divided  into  any  number  of 
equal  parts,  and  suppose  M^  and  M.^  two  of  the  dividing  points. 
Erect  corresponding  ordinates  at  these  points,  and  at  their  ex- 


J4S  ANALYTIC    GEOMETRY. 

tremities  draw  parallels  to  ^^'^4,  forming  pairs  of  corresponding 
rectan;;ks,  of  which  J/,  ^  and  J/,  ^'  are  one  pair. 
Now  from  the  figure  we  see  that 


M^R^  ~  M^M^  X  yl  ~  //  "  a 

Sinnilarly  we  may  show  that  the  area  of  any  corresponding 
rectangles  of  ellipse  and  circle  are  in  the  ratio  h  -.  a ;  therefore 
the  sum  of  the  rectangles  of  the  ellipse  is  to  the  corresponding 
sum  for  the  drcle  as  ^  :  dr.  Now  if  we  increase  indefinitely  the 
number  of  equal  divisions  of  A'  A,  the  sum  of  the  rectangles  of 
the  ellipse  approaches  the  \\7iM'ZXt2i  of  the  ellipse  as  a  limit,  and 
the  rectangles  of  the  circle  approach  the  half-area  of  the  drcle 
as  a  limit ;  therefore,  by  the  ITieory  of  Limits,  the  area  of  the 
ellipse  is  to  the  area  of  the  circle  as  ^  :  <at.     Since  the  area  of 

the  circle  is  tt  a^,  the  area  of  the  ellipse  must  be  tt  ix^  ~,  or  ira  ^. 

EXAMPLES, 
(i.)    Find  the  area  of  the  ellipse  whose  equation  is 

^^  -  4>'^  ~  loo. 
(2.)    Find  the  ratio  of  the  areas  of  the  ellipses 
rjx^  +  16  y=  144, 

4^^+    9/  ^  144- 

^;2j.   2:3:4. 

86.  J.ct  us  find  the  equation  of  an  hyperbola  whose  foci  lie- 
on  the  axis  of  r,  whose  transverse  axis  is  2  <^,  and  whose  conju- 
gate axis  is  2  ^/. 

We  will  first  find  the  equation  of  an  hyperbola,  with  its  foci  on 
O  V,  wi  h  2  </  forks  transverse,  and  2^  for  its  conjugate,  axis. 
J  ;>i .  iri  ly  be  (AA'dlned  from  the  cqwdUon  of  §  64,  by  turning  the 


•nil',  CONIC  sKC'iioNS.  149 

axes  through  an  an^lc  of  90".  Usin;^  tlic  forniuhis  for  Iraiis- 
formation  given  in  §  42,  (Jor.,  we  have  for  tiie  equali(jn  of  this 
hyi)erl)ohi  ,      ^2 

-,-77  =  1 (I) 

The  transverse  axis  of  the  recjuired  hyperl)ohi  has  tin;  same 
length  as  the  conjugate  axis  of  this  one,  and  lucc  iwrsa ;  (luM-e- 
fore  we  shall  obtain  the  required  ecjuaticMi  by  intercliaii;;iii;;  a 
and  b  in  (i),     Tliis  gives 

which  must  be  the  equation  sought. 

This  hyperbola  is  said  to  l)e  conjugate  lo  llic  one  wIios(;  (-(jua- 
tion  was  found,  and  llic  two  are  closely  connected.  We  may 
therefore  delinc;  lonju^atc  hyperbolas  as  lliose  which  have  the 
same  axes,  the  transverse  axis  of  one  being  the  conjugate  axis 
of  the  other,  and  the  conjugate  axis  of  the  one  the  transverse 
axis  of  the  other. 

EXAMPLES, 
(i.)     I'uid  the  ecjuation  of  the  hyperbola  conjugate  to 

X'  -  4/   =z   36, 

and  determine  its  foci,  axes,  and  eccentricity. 

Ans.    4/      x'  rz:  36,    (o,  1  3  V.S),  3,  6,  Vs, 
are  the  eqiialion,  foci,  semi  transverse   axis,  semi-conjugate  axis, 
and  eccentricity,  res[)ectively. 

(2.)    Find  the  equations  of  tangent  and  normal   Xo  the  hyper 
bola  conjugate  to 

x^  _  y'  _ 

a"    ~   b'  ~  '' 
at  a  point  /\  of  the  curve 

{.Vxy      x^x 
Ans.     y^^    "a^-^  '    '''"^ 

(b'x,y  +  a'y,  x  =  {a""  +  b^)  x,  y^. 


ISO 


ANALYTIC    GEOMETRY. 


87.   C 071  jugate  hyperbolas  have  the  same  asymptotes. 

For,  turning  the  axes  through  90°,  and  interchanging  a  and  b 

the  equations  of  the  asymptotes  to  the  hyperbola 


a' 


found  in  §  70,  we  have 
X  = 

for  the  equations  of  the  asymptotes  to  the  hyperbola 


a  ,  b 

x  =  ±  -y,    or  y  =  ±-x, 


(0 


21 


Comparing  (i)  with  the  equations  of  the  asymptotes  in  §  70 
we  see  that  the  equations  are  alike,  which  proves  our  proposi- 
tion. 

88.  In  the  accompanying  figure,we  showa  pair  of  conjugate  liy- 
perbolas    with 
their   common 
asymptotes  6^Z 
and  O  L'. 

A'  A  and 
B^  B  are  axes 
of  both  curves; 
the  one  with 
A^  A  for  its 
transverse  axis 

has  foci  7^  and  F^ ;  the  one  which  has  B  B'  for  its  transverse 
axis  has  foci  F^  and  F^ . 

The  reader  may  easily  deduce  the  following  properties  from 
those  already  proved  and  the  definitions  which  have  been  given  : 

The  four  foci  lie  on  the  circumference  of  a  circle  whose  radius 
is  c. 

The  rectangle  which  has  its  sides  parallel  and  equal  to  the 
axes,  and  the  common  centre  of  the  hyperbolas  for  its  centre, 


^ 

%^ 

F, 

^^ 

^' 

\ 

^"--^ 

i      \^ 

/ 

1 

A'     ^^ 

b' 

A 

,F 

\ 

^ 

F^' 

X. 

THE    CONIC    SECTIONS.  I51 

lies  wholly  between  the  curves,  having  each  side  tangent  to  one 
of  the  branches  of  these  curves,  and  the  asymptotes  of  the 
hyperbolas  for  its  diagonals. 

An  ellipse  with  the  same  axes  as  the  hyperbolas  will  be  in- 
scribed in  this  rectangle  and  will  touch  the  hyperbolas  at  A,  A' , 

89.  The  definitions  of  the  conic  sections  which  we  gave  at 
the  beginning  of  this  chapter  fail  to  bring  out  the  fact,  which 
appears  on  closer  study,  that  these  three  curves  are  merely 
different  species  of  a  single  genus.  It  will  therefore  be  of  in- 
terest to  give  a  new  definition  which  covers  equally  the  three 
cases  of  the  ellipse,  the  hyperbola,  and  the  parabola.  We  shall 
in  this  way  gain  a  more  comprehensive  view  of  these  curves 
and  learn  some  new  facts  about  them. 

Boscovich's  Definition. — A  Conic  Section  is  the  locus 
of  a  point  which  moves  in  a  plane  so  that  its  distance  from  a 
fixed  point  of  the  plane  bears  a  constant  ratio  to  its  distance 
from  a  fixed  line  in  the  plane. 

The  fixed  point  is  called  \.\\q  focus,  the  fixed  line  iht  direc- 
trix, and  the  value  of  the  constant  ratio  the  eccefitrtcity  of  the 
conic  section. 

We  will  now  examine  this  definition  and  show  that  it  is  not 
in  contradiction  with  the  definitions  previously  laid  down. 

Let  us  take  the  directrix  in  the  above  definition  as  axis  of 
Y  and  the  perpendicular  dropped  upon  it  from  the  focus  F  as 
axis  of  X.  We  will  call  the  eccentricity  c,  and  the  distance 
from  the  directrix  to  the  focus  ;;/,  so  that  the  coordinates  of  F 
are  (w,  o).  Denoting  by  {x,y)  the  coordinates  of  the  moving 
point  P  which  traces  out  the  locus,  we  obtain  at  once  as  the 
equation  which  expresses  the  definition  of  the  curve 


V{x-  mf  +7^ 


(i) 


^WrVEF^SITY  OF  CALJF01c,NiA 


152  ANALYTIC    GEOMETRY. 

This  equation,  when  simplified,  takes  the  form 

(i  —  e^)x^  f  y  —  2mx  -\-  m^  =  o (2) 

This  is  the  equation  of  the  conic  section. 

If  (?  =  I  this  equation  reduces  to  equation  (3),  §  65,  and  in 
fact  it  will  be  seen  that  in  this  case  our  present  definition 
of  a  conic  section  is  the  same  as  our  old  definition  of  a 
parabola,  and  our  use  of  the  terms  focus  and  directrix  coin- 
cide in  these  cases.  We  have  merely  introduced  the  new  term 
eccentricity,  which  up  to  this  time  we  had  not  used  in  the  case 
of  the  parabola,  in  such  a  way  that  the  eccentricity  of  every 
parabola  is  i. 

Considering  now  the  cases  where ^  ^  i,let  us  find  the  inter- 
cepts of  our  curve  on  the  axis  of  X.  This  is  done  by  letting 
^  =  o  in  (2): 

( I  —  <f')  x^  —  2  in  X  +  w'  =  o. 

The    roots    of    this  equation    are    — — — .       Accordingly  the 
curve  (2)  cuts  the  axis  of  X  in  the  two  points  A,  B: 

Denoting  the  distance  between  these  two  points  by  2  a  we 
have 

em  ,  X 

"  =  i~--.' (''^ 

In  order  to  simplify  equation  (2)  let  us  now  transform  ft  to 
a  new  system  of  rectangular  coordinates  parallel  to  the 
old,  with  origin  half  way  between  the  points  (3),  i.e.  at  the 
point  C: 


(7^-4 


THE    CONIC    SECTIONS.  153 

The  formulas  for  this  transformation  are  (v.  (3),  §  41): 

Performing  this  transformation  and  dropping   accents,  equa- 
tion (2)  becomes 

If  we  introduce  the  quantity  a  by  means  of  (4)  this  equa- 
tion takes  the  form 

a^  +  a^r    -  .')  =  ' (s) 

Let  us  examine  first  the  case  e  <  1.     In  this  case  (5)  repre- 
sents an  ellipse  whose  semi-axes  are  a  and 

€  m 


b  =  aVi  -."=--^^== (6) 

y  I  —  <f 

this  latter  being  obviously  the  semi-minor  axis. 
Moreover,  we  have 


m  em 


FC=OC-OF  =  —^^^,-m  =  -^-^=ae^  Va'-b\  (7) 
1  —  e  1  —  e^ 

Accordingly  F,  which  we  have  called  the  focus  of  the  conic, 
is  really  one  of  the  foci  of  the  ellipse  according  to  our  earlier 
definition. 

Finally  the  eccentricity  of  the  ellipse  was  defined  on  p.  122 
as 

c^_FC  _^_ 
a        a         a  ^ 

so  that  our  new  definition  of  the  eccentricity  coincides  with 
the  old. 


154  ANALYTIC    GEOMETRY. 

The  conception  of  a  directrix  for  the  ellipse  is  something 
we  did  not  have  before.  Since  the  ellipse  is  symmetrical  with 
regard  to  the  minor  axis,  it  is  clear  that,  besides  the  focus  and 
directrix  with  which  we  started,  there  is  a  second  focus  and 
a  second  directrix  situated  the  same  distance  to  the  right 
of  the  minor  axis  as  the  first  were  to  the  left.  The  distance 
Trom  the  directrix  to  the  minor  axis  is 


I  —  e        e 


Thus,  when  the  centre  of  the  ellipse  is  taken  as  origin,  our 
two  directrices  are 

*=±7 (8) 

e 

Turning  now  to  the  case  <r  >  i  we  see  that  (5)  represents 
a  hyperbola  whose  semi-transverse  axis  is  a,  while  its  semi- 
conjugate  axis  is 

b  =  a  VT^^i (9) 


Moreover,  we  have 


em 


FC==OC-  OJ^= ^=ae=   Va' -^  d\  .    (10) 

I  —  e 

so  that  F  is  really  a  focus  of  the  hyperbola  according  to  our 
earlier  definition. 

As  in  the  case  of  the  ellipse  we  see  that  our  new  definition 
of  the  eccentricity  coincides  with  the  old. 

We  also  see  that  the  hyperbola  has  two  directrices  whose 
equations  are  (8). 

The  ellipse  and  hyperbola  are  sometimes  called  the  central 
conies  in  distinction  to  the  parabola,  which  has  no  centre. 


THE    CONIC    SECTIONS.  155 

90.  From  the  point  of  view  of  the  last  section  the  three 
kinds  of  conic  sections  are  distinguished  from  one  another  by 
the  value  of  the  eccentricity,  and  we  have 

An  Ellipse  when  <f  <  i, 
A  Parabola  when  <?  =  i, 
An  Hyperbola  when  e  >  i. 

Thus  we  see  that  the  parabola  may  be  regarded  as  a  form  of 
curve  intermediate  between  the  ellipse  and  the  hyperbola.  If 
in  equation  (2)  we  keep  m,  the  distance  from  the  directrix  to 
the  focus,  unchanged  and  allow  e,  starting  with  a  value  less 
than  I,  to  increase,  we  have  an  ellipse  whose  major  axis 
(v.  (4))  and  whose  minor  axis  (v.  (6))  are  both  increasing. 
If  we  allow  <r  to  increase  towards  the  value  i  as  its  limit, 
tnese  axes  increase  indefinitely,  and  the  ellipse  approaches  a 
parabola  as  its  limiting  form.  On  the  other  hand,  if  e,  starting 
with  a  value  greater  than  i,  decreases  towards  i  as  its  limit, 
the  axes  of  the  hyperbola  represented  by  (2)  increase  indefi- 
nitely. One  branch  of  the  hyperbola,  therefore — the  left- 
hand  one  if  m  is  positive — moves  farther  and  farther  away, 
while  the  other  branch  approaches  a  parabola  as  its  limiting 
form. 

91.  It  can  easily  be  seen  that  every  conic  except  a  circle 
can  be  obtained  by  the  method  of  §  89.  This  is  clearly  so  for 
parabolas,  since  in  this  case  the  method  of  generating  the  conic 
given  in  §  89  is  practically  the  same  as  that  given  in  §  62. 
In  the  case  of  an  ellipse  or  hyperbola  the  conic  is  completely 
determined  by  its  semi-axes  a  and  b.  These  in  turn  deter- 
mine the  eccentricity  e,  and  e  and  a  determine  w  by  (4) 
except  in  the  case  of  a  circle,  for  which  e  =  o,  while  a  is  in 
general  not  zero,  so  that  (4)  involves  a  contradiction.  The 
fact  that  the  circle  is  not  included  under  Boscovich's  defi- 
nition of  a  conic  is  easily  understood  when   we  notice  that 


156  ANALYTIC    GEOMETRY. 

the  circle  is  the  limiting  form  of  an  ellipse  as  the  eccentricity 
approaches  zero.  Accordingly  by  (8),  §  89,  ihe  directrices  of 
the  ellipse  move  off  indefinitely,  so  that  the  circle,  having 
no  directrices,  cannot  come  under  a  definition  which  involves 
the  use  of  a  directrix. 

EXAMPLES. 

(i.)  Find  the  equation  of  the  tangent  to  the  conic  (2)  at 
the  point  {-^\,yj)- 

(2.)  Prove  that  tangents  drawn  to  any  conic  at  the  extremi- 
ties of  a  focal  chord  intersect  on  the  directrix. 

(3.)  A  point  moves  so  that  its  distance  from  a  fixed  point  is 
always  equal  to  its  distance  from  a  fixed  circle.  Prove  that 
its  locus  is  a  conic  having  the  fixed  point  as  focus. 

Remark. — The  fixed  circle  of  this  example  is  sometimes 
called  the  director  circle  of  the  conic. 

(4.)  Show  that  every  central  conic  has  two  director  circles 
(v.  Ex.  3),  and  find  their  equations. 


CHAPTER   VIII. 
DIAMETERS.     POLES   AND   POLARS. 

92.  According  to  the  definition  of  p.  89,  the  diameter  of  a 
conic  section  is  the  locus  of  the  middle  points  of  a  set  of  par- 
allel chords. 

Beginning  with  the  ellipse,  let  us  find  the  locus  of  the  middle 
points  of  a  set  of  parallel  chords  whose  equations  are 

y  =  ^,x  +  b, (i) 

y  =  X^x  +b^ (2) 

etc. 

The  extremities  of  the  first  chord  may  be  found  by  combining 
its  equation  with  that  of  the  ellipse. 

Using  the  form  of  the  equation  given  in  §  6t^,  equation  (6), 
and  substituting  for^  its  value  from  equation  (i),  we  have 

(^2  4-  a^  A/)  A-^  +  2a'b^  A,  .T  +  a-"  (^,^-  b')  =  o. 

The  roots  of  this  equation  are  the  abscissas  of  the  extremities 
of  the  chord,  and  therefore,  by  §  11,  and  the  foot-note  on  p.  90, 
the  abscissa  x'  of  the  middle  point  of  the  chord  is 

"^  ~     b^  +  a'x;^ ^^^ 

The  coordinates  of  F' y  the  middle  point  of  chord  (i),  must 
of  course  satisfy  equation  (i),  and  therefore 

which  becomes,  after  reducing, 


,  _      b'b, 

^  -  VT~a^- ^'♦^ 

157 


15^  ANALYTIC    GEOMETRY. 

Dividing  y'  by  x',  we  obtain  from  (4)  and  (3)  a  necessary  re- 
lation between  them  which  is  independent  of  /^i,  the  intercept  of 
the  first  chord  on  the  axis  of  y.     It  is 

•4=-4 (5) 

x'  a"  \ 

This  relation  must  exist  between  the  coordinates  of  the  middle 
points  of  each  of  the  set  of  parallel  chords,  and  is  therefore  the 
equation  of  a  diameter.     Dropping  the  accents  we  may  write  it 

■^=-.-5:-' (^> 

which  is  in  the  form 
where 

^=-4 (7) 

From  the  form  of  the  equation,  we  know  that  the  diameter 
passes  through  the  origin  or  centre ,  but  it  is  not,  as  for  the  cir- 
cle, perpendicular  to  the  chords  which  it  bisects,  for  the  slopes  of 
chord  and  diameter  do  not  satisfy  the  test  of  perpendicularity. 

It  may  be  shown  in  the  same  manner  as  for  the  circle 
(v.  §  52)  that  every  chord  through  the  centre  is  a  diameter. 

93-  Using  \  to  represent  the  slope  of  that  diameter  which 
bisects  chords  with  a  slope  \,  let  us  find  the  slope  of  the  diame- 
ter which  bisects  all  chords  whose  slopes  are  A^. 

Calling  the  required  slope  A,  we  have  by  equation  (7)  of  the 
last  section 

'=-^; ^'^ 

where 

Substituting  this  value  of  Ag  in  (i),  we  have 


DIAMETERS.       POLES    AND    POLARS. 


159 


^  =  K (2) 

the  slope  of  the  original  set  of  chords. 

Therefore,  if  one  of  two  diameters  bisects  chords  parallel  to  tht 
other  ^  the  second  will  also  bisect  all  chords  parallel  to  the  first.    Such 

diameters   are  called   co7iju- 
gate. 

In  the  figure,  P^  P^  and 
P^  P^  represent  a  pair  of 
conjugate  diameters  of  the 
ellipse,  and  each  of  these 
lines  will  bisect  all  chords 
parallel  to  the  other. 

The  product  of  the  slopes 
of  a  pair  of  conjugate  dia- 
meters is  constant;  for  we  readily  obtain  from  equations  (i) 
and  (2) 


\\  =  -r. 


(3) 


Note.     The  results  of  this  section  may  be  obtained  at  once 
from  equation  (7)  of  the  last  section  by  writing  it  in  the  form 

b"" 


\  = 


where  Xj  bears  the  same  relation  to  A  which,  in  the  original  form, 
X  bore  to  \.  From  this  we  may  draw  the  inferences  stated  in 
this  section. 

Equation  (3)  shows  that  the  product  of  the  slopes  of  conjugate 
diameters  is  constantly  negative ;  therefore  one  of  these  slopes 
must  be  positive  and  the  other  negative.  It  follows  that  the  in- 
clination of  one  of  the  diameters  must  be  between  0°  and  90°, 
and  of  the  other  between  90°  and  180°,  except  in  the  special 
case  where  the  diameters  are  the  axes  of  the  ellipse,  (v,  §  25, 
Rem.) 


6o 


ANALYTIC    GEOMETRY. 


Therefore  conjugate  diameters  of  an  ellipse  lie  on  opposite  sides  oj 
the  minor  axis. 

94.  Changing  the  sign  of  IP-  in  the  equations  of  the  last  two 
sections,  we  have  for  the  equation  of  a  diameter  of  the  hyperbola 
bisecting  chords  with  a  slope  \. 


a^\ 


X  : 


and  for  the  product  of  the  slopes  of  conjugate  diameters 


^1  ^2  =  -2* 


(i) 


(2) 


Let  the  student  verify  these  equations  by  finding  the  equation 
of  a  diameter  of  an  hyperbola,  and  show  that  if  one  diameter  of 
an  hyperbola  bisects  all  chords  parallel  to  the  other,  the  second 
bisects  all  chords  parallel  to  the  first. 

In  the  figure  P^  P^  and  P^  P^  represent  a  pair  of  conjugate 
diameters  ;  P^  P^  extended 
bisects  all  chords  of  either 
branch  of  the  hyperbola, 
which  are  parallel  to  P^  P^; 
and  P.l  P^  extended  bisects 
all  chords  lying  between  the 
two  branches  of  the  curve 
and  parallel  to  P^  P^.  We 
limit  Pc^  P^  by  the  conjugate  hyperbola,  as  will  be  explained 
later. 

It  follows  from  equation  (2)  that  conjugate  diameters  of  an 
hyperbola  lie  on  the  same  side  of  the  conjugate  axis,  as  shown  in  the 
figure. 

Also,  the  numerical  value  of  the  slope  of  one  diameter  must  be 

less  than  -,  and  of  the  other  greater  than  -,  except  in  the  case 


DIAMETERS.       POLES    AND    POLARS.  l6l 

where  each  equals  —  and  the  diameters  coincide  with  one  of  the 
^         a 

asymptotes. 

We  learn  from  this  that  one  diameter  has  a  less  inclination 
than  the  asymptote,  and  the  other  a  greater;  therefore  one  lies 
in  the  angle  Z'  O  Z,  and  the  other  in  the  supplementary  angle. 
We  may  also  state  this  fact  as  follows  :  — 

Of  two  conjugate  diameters  of  an  hyperbola,  oJily  one  meets  the 
hyperbola,  the  other  meeting  the  conjugate  hyperbola. 

EXAMPLES. 
(i.)    Find  the  equation  of  the  diameter  of  the  ellipse 

X-       y^ 

which  bisects  the  chord  whose  equation  is 

3  a:  +  _>'  +  2  =  o. 

Ans.    3  ^  —  i6  j^  =  o. 

(2.)    What  is  the  equation  of  the  diameter  conjugate  to  the 

one  found  in  the  last  example  ? 

Ans.   3  X  -\-y  —  o. 

(3.)  Find  the  equations  of  a  pair  of  conjugate  diameters  of 
the  hyperbola 

x^  —  12  y^  =  48, 

one  of  which  bisects  the  chord  whose  equation  is 
2^-9  J  =  3. 

Ans.   ^x  T^S  y   and    2  x  =  g  y. 
(4.)    Find  the  equation  of  the  diameter  of  the  ellipse 
4x^  +  gy^  =  144, 

which  has  for  one  extremity  the  point  (3  Vs,  2) ;  what  is  the 
other  extremity  of  this  diameter  ? 

Ans.    2x  =  ^Vsy,    (  -  3  V3»  -  2). 


1 62  ANALYTIC    GEOMETRY. 

(5.)    Find  the  equation  and  extremities  of  the  diameter  con- 
jugate to  the  one  found  in  the  last  example. 

Ans.   2x  +  V3y  =  o,  (3.-2  V3)  (  -3)  2  V^). 
(6.)   One  extremity  of  the  diameter  of  the  ellipse 


,2  +-72-  =  ! 


is  the  point  {x^,  y^  ;  what  is  the  other  extremity  ?   what  the 

equation  of  the  diameter  ? 

Ans.    (-x^,  -y,),y^x  =  x,y. 

(7.)   What  is  the  equation  of  the  diameter  conjugate  to  the 

one  found  in  the  last  example  ? 

.       x,x     y^y 

(8.)  Remembering  that  (x^,  y^)  is  a  point  of  the  curve,  prove 
that  the  extremities  of  the  diameter  of  Ex.  7  are  the  points 

Oi.-^^.)   and   (-f/p^^i). 

(9.)  Show  that  the  tangent  to  an  ellipse  at  any  point  of  the 
curve  is  parallel  to  the  diameter  conjugate  to  the  one  which 
passes  through  that  point ;  and  that  tangents  at  opposite  extrem- 
ities of  any  diameter  are  parallel.  Prove  the  same  property  for 
the  hyperbola. 

(10.)  Find  the  equation  of  the  diameter  of  the  hyperbola 
x^       y^ 

which  meets  the  curve  in  the  point  -Pj ;  also,  the  equation  of  the 
conjugate  diameter. 

X,  X     y^y 
Ans.  y,x-x,y=o,    -^^ ^  =  °- 

(11.)    Show  that  the  coordinates  of  the  points  where  the  sec- 


DIAMETERS.       POLES    AND    POLARS.  163 

ond  diameter  meets  the  hyperbola  are  imaginary,  their  values 
being 

(12.)    Show   that  the  second  diameter  meets  the  hyperbola 
conjugate  to  the  one  given  in  example  (lo)  in  the  points 


(a        b      \     (    a  b      \ 


(13.)  Find  the  equation  of  the  diameter  which  bisects  all 
chords  with  a  slope  A^  in  the  hyperbola  conjugate  to 

and  show  that  it  coincides  with  the  corresponding  diameter  of 
this  hyperbola. 

(14.)  Show  that  in  the  figure  of  §  94,  /*/  P^  and  P/  ^2  ^^^  ^ 
pair  of  conjugate  diameters  for  each  of  the  two  conjugate  hyper- 
bolas. 

(15.)  Find  the  equation  of  the  diameter  of  a  parabola,  which 
bisects  all  chords  whose  slope  is  \ ;  show  that  all  diameters  of  a 

parabola  are  parallel  to  the  axis. 

m 
Ans.  y  =  —. 
1 

(16.)    Prove  that  when  a  parabola  lies  on  the  right  oi  O  Y 

(i.  e.  when  in  is  positive),  the  diameter  bisecting  parallel  chords 

whose  inclination   is  less   than  90°   lies  above   the  axis  of  the 

curve  ;  the  diameter  bisecting  chords  whose  inclination  is  greater 

than  90°  lies  below  the  axis.     State  the  corresponding  theorem 

when  ;;/  is  negative.     What  is  the  diameter  which  bisects  chords 

perpendicular  to  the  axis  ? 

95-  By  means  of  the  results  of  example  (8)  of  the  last  section 
we  can  find  the  lengths  of  a  pair  of  conjugate  diameters  of  an 


l64  ANALYTIC    GEOMETRY. 

ellipse  in  terms  of  the  coordinates  of  an  extremity  P^  of  one  of 
them.  For,  since  the  centre  is  the  middle  point  of  each  diam- 
eter, if  we  let  a^  and  b^  represent  the  halves  of  the  conjugate 
diameters,  we  have 

a^^  =  x:^^y,\. (I) 

and 


a 


V" 


'"--wy'-^^'^' (^) 


Now  F^  is  a  point  on  the  ellipse,  and  therefore 


yi'  =  ^,  (a'  -  X,') (3) 

Substituting  this  value  in  equations  (i)  and  (2),  we  have 

a''^  =  X,' +  d^'  -  ^^x,' 

=  b'  +  e'  x,\ (4> 

and 

^d'-e'x^ (5J 

Equations  (4)  and  (5)  give  the  squares  of  the  half-diameteis 
in  terms  of  .Tj,  y^^  and  quantities  which  are  know^n  when  the 
ellipse  is  given,  and  from  them  the  lengths  of  the  diameters  may 
be  found. 

96.  For  the  hyperbola,  one  of  the  semi-diameters  is  imaginary 
and  the  expression  for  its  length  contains  the  factor  V  —  i,  for 
we  have  shown  in  §  94>  Ex.  11,  that  only  one  of  a  pair  of  coniu- 
gate  diameters  meets  the  hyperbola  in  real  points.  Using  <?'  to 
represent  the  real  semi-diameter,  and  //  to  represent  the  real 
factor  in  the  expression  for  the  imaginary  semi-diameter,  —  a 


DIAMETERS.       POLES    AND    POLARS.  165 

use  analogous  to  that  of  b  (v.  §  69), — we  may  write  for  the 
hyperbola 

a}''^e'  x^  ~b\ (i) 

and 

b^^'^e'  x^  -a" (2) 

EXAMPLES. 

(i.)  Prove  that  the  sum  of  the  squares  of  any  two  conjugate 
semi-diameters  of  an  ellipse  is  constant  and  equal  to  the  sum 
of  the  squares  on  the  semi-axes. 

(2.)  State  and  prove  the  theorem  for  the  hyperbola  corre- 
sponding to  that  given  in  the  last  example  for  the  ellipse. 

(3.)  Prove  that  conjugate  diameters  of  an  equilateral  hyper- 
bola are  equal. 

(4.)  Show  that  the  ellipse  has  a  pair  of  equal  conjugate 
diameters,  and  show  that  they  coincide  with  the  asymptotes  of 
an  hyperbola  which  has  the  same  axes  as  the  ellipse. 

(5.)  Prove  that  the  equal  conjugate  diameters  of  an  ellipse 
bisect  the  lines  joining  the  extremities  of  the  axes. 

(6.)  Show  that  the  perpendicular  from  the  centre  of  an  el- 
lipse upon  a  tangent  at  a  point  P^  of  the  curve  is  equal  to  -jj^ 

where  b^  is  the  semi-diameter  conjugate  to  the  one  which  passes 
through  Py 

(7.)  Prove  that  all  parallelograms  circumscribing  a  given 
ellipse,  and  having  their  sides  parallel  to  a  pair  of  conjugate 
diameters,  have  the  same  area. 

(8.)  Find  the  equations  of  the  chords  of  an  ellipse,  which 
connect  the  extremities  of  a  pair  of  conjugate  diameters,  one  of 
which  meets  the  curve  in  P^ 

Ans.  b  (J?  x^  —  a  y^  X  +  a  {a  y^  -\-  b  x^  y  —  a^  b"^  =  o 
is  the  equation  of  one  of  the  chords. 


l66  ANALYTIC    GEOMETRY. 

(9.)  Find  the  locus  of  the  middle  point  of  the  chord  of  the 
last  example  as  the  point  P ^  moves  around  the  ellipse. 

97.  Two  points  C  and  D  are  said  to  divide  the  straight  line 
A  B  harmonically  if  one  of  them  divides  this  line  internally  in 
the  same  ratio  as  the  other  divides  it  externally  ;  />  ,  having  re- 
gard to  sign,  if 

CA_DA  ,  . 

~CB~       DB ^'^ 

This  equation  may  also  be  written 

AC  BC 


AD  B  D 


(2) 


The  left-hand  side   of   (2)  is   the   ratio  in  which  the  point  A 

divides  the  line  CD,  while  the 

•4 1 i ! '     right-hand  side   is  the   negative 

of  the  ratio  in  which  B  divides 
this  line.  Accordingly  (2)  expresses  the  fact  that  A  and  B 
divide  the  line  CD  harmonically.     Hence  the  theorem  : 

If  the  points  C,  D  divide  the  line  A  B  harmonically^  then  con- 
versely the  poi?its  A,  B  divide  the  line  CD  harmonically, 

98.  If  the  three  points  A,  B,  C  are  given,  we  can  in  general 
construct  a  fourth  point  D  so  that  C,  D  divide  A  B  harmoni- 
cally. This  is  evident  from  Plane  Geometry,  since  if  C  divides 
A  B  internally  all  we  have  to  do  is  to  construct  the  point  di- 
viding this  line  externally  in  the  same  ratio^  and  vice  versa. 
Analytically  we  can  see  the  same  fact  as  follows.  Let  the 
coordinates  of  A  be  {x, ,  )\),  of  B^  {x^,y^,  and  let  m^  :  m^  be 
the  ratio  in  which  C  divides  A  B.  Then  (v.  p.  9)  the  coordi- 
nates of  C  are 

hn^x^  —  m,x^        m,y,  —  fn,y\  ^  ^      ^  , 

\      7n,  —  ?n.       '  m^  —  m,    I 


DIAMETERS.       POLES    AND    POLARS.  1 67 

and  the  point  D  which  divides  A  B  \n  the  latio  —  m^  :  m^  has 
the  coordinates 

\      m^  +  in^      '  m^  +  7n^      j ^  ^ 

We  thus  see  that  there  is  no  difficulty  in  finding  the  point  D 
except  in  the  one  case  in  which  w,  =  —  tn^',  i.e.^  when  C bisects 
the  line  A  B.  In  this  case  it  is  also  obvious  geometrically 
that  the  point  D  cannot  be  constructed,  since  every  point  on 
the  line  A  B  produced  is  nearer  to  one  end  than  to  the  other. 
It  is  often  more  convenient  to  use  a  single  letter  to  denote 
the  ratio  in  which  C  divides  A  B.     If  we  write 

the  formulae  (i)  and  (2)  for  the  coordinates  of  C  and  £>  take 
the  form 

V  I  -  /^  -'     I-/.  ] (2) 

\"TT7^'      1.  ^  fx]  '  "  ' ^4) 

99.  By  the  />o/ar  of  a  fixed  point  B  with  regard  to  a  given 
conic  is  meant  the  locus  determined  as  follows.  Through  B  a 
straight  line  is  passed  cutting  the  conic  in  C  and  B>.  On  this 
line  a  fourth  point  Q  is  constructed  so  that  B,  Q  divide  C  B> 
harmonically.  The  line  B  C  B>  is  then  allowed  to  revolve 
about  B,  the  point  Q  being  constructed  on  every  position  of 
the  line  in  the  manner  just  indicated.  The  locus  of  Q,  which 
we  shall  prove  to  be  in  all  cases  a  straight  line  iS  called  the 
polar  of  B. 

Let  us  consider  first  the  case  in  which  the  coriic  is  an  ellipse. 


i68 


ANALYTIC    GEOMETRY. 


As  usual  we  will  use  the  major  and  minor  axes  of  this   ellipse 
as  coordinatvf  axes,  writing  its  equation  in  the  form 


-H  a'y'  =  a'b\ 


(i) 

The  coordinates  of  P  shall  be  [x^.y').  Call  the  coordinates 
of  the  moving  point  Q,  whose  locus  we  want  to  find  {x\y'). 
Since  by  hypothesis  F^  Q  divide  C  Z>  harmonically,  conversely 


C,  D  must  divide  F  Q  harmonically  (v.  §  97).  We  will  intro- 
duce as  an  auxiliary  variable  (v.  §  56)  the  ratio  in  which  the 
point  C  divides  the  line  F  Q.  Calling  this  ratio  //,  we  have  by 
formulae  (3)  and  (4)  of  §  98  as  the  coordinates  of  C  and  D 
respectively 

(x,-^ix'      y,  -  ^y\  , 

/x^  /Ax^      y,  +  ^iy'\  ,. 

V    I  +  /(     '        I  +  yw  / ^^^ 


DIAMETERS.       POLES    AND    POLARS.  169 

Since  C  and  D  lie  on  the  ellipse,  we  have  the  following  two 
equations  obtained  by  substituting  the  coordinates  (2)  and  (3) 
in  (i)  and  then  clearing  of  fractions  : 

H'ix,  -  )i  x'Y  +  a^  (/.  -  l^y'Y  =  a-f(i-iAY  .  .  (4) 

nx,  +  ^ix'Y  +  a^  (y,  +  M/r  =  «'^'  (I  +  My . .  (s) 

Between  these  two  equations  we  must  eliminate  /a.  This  can 
be  done  by  subtracting  (4)  from  (5),  getting 

2//  i^'  X,  x'  -^  2  }Aa  ^r,  y'  =  2  }.ia''b'' (6) 

The  factor  2  yw  can  be  cancelled  out  from  this  equation,  and 
we  then  have  as  the  et^uation  of  the  polar,  after  dropping  the 

accents, 

b'x.x  +  a'yj^a'b' (7) 

or 

This  equation,  being  of  the  first  degree,  represents  a  straight 
line. 

Replacing  b"^  by  —  b'^,  we    find    as   the   polar  of  the  point 
{x^,y^)  with  regard  to  the  hyperbola 

-^a"-  ¥  =  ^^     (9) 

the  straight  line 

~a^-l^  =  ^ (^°^ 

Show    by    the    same    method    that   the  polar  of    the  point 
(^lO'i)  ^'it^^  regard  to  the  parabola 

y  =  2  /;/  jc (11) 

is  the  straight  line 

y^y  —  ni  {x  +  x^) (12) 


I70  ANALYTIC    GEOMETRY. 

If  a  =  l>f  the  ellipse  (i)  becomes  the  circle 

^'+/  =  a^ (13) 

Accordingly  the  polar  of  (:v„  y^)    with  regard   to   this  circle  is 
the  straight  line 

x,x+y,y=a'' (14) 

EXERCISES, 
(i.)  Find  the  polar  of  the  point  (2,  —  i)  with  regard 

(a)  to  the  ellipse  —  +  <-  =  i  ; 

9        4 
(^)    to  the  hyperbola  2  x''  —  3^  =  i  ; 
(c)    to  the  parabola  j'^  =  6  x. 

(2.)  Prove  that  the  polar  of  a  point  with  regard  to  a  circle 
is  perpendicular  to  the  line  joining  the  point  to  the  centre  of 
the  circle  ;  and  that  the  radius  of  the  circle  is  a  mean  propor- 
tional between  the  distances  of  the  point  and  its  polar  from  the 
centre  of  the  circle. 

(3.)  Find  the  equation  of  the  polar  of  the  point  {x^,y^)  with 
regard  to  the  hyperbola  conjugate  to  (9). 

Hence  prove  that  the  polars  of  a  point  with  regard  to  two 
conjugate  hyperbolas  are  parallel.  What  more  can  you  say 
about  the  relative  position  of  these  two  lines  ? 

(4.)  Find  the  equation  of  the  polar  of  the  point  (x^,y^) 
with  regard  to  the  conic  (2)  of  §  89. 

Hence  show  that  the  polar  of  a  focus  of  a  conic  is  the  cor- 
responding directrix. 

100.  It  will  be  noticed  that  fornuike  (8),(ro),  (12),  and  (14) 
for  the  polars  of  (x,,  y^)  with  regard  to  the  ellipse,  hyperbola, 
parabola,  and  circle  are  identical  with  the  formulae  we  obtained 
in  the  last  chapter  (§§  74,  75,  76)  and  in  §  49  for  the  tangents 


DIAMETERS.       POLES    AND    POLARS. 


171 


to  these  curves  at  the  point  (x^,yj.  The  difference  is  that  in 
finding  the  tangent  we  assumed  that  (^,,  y^)  lies  on  the  curve, 
while  we  have  now  assumed  that  (^i,J,)  does  not  lie  on 
the  curve.  In  fact  the  definition  of  the  polar  which  we  have 
given  obviously  breaks  down  if  jR  lies  on  the  conic.  It  is  cus- 
tomary, however,  in  this  case  to  speak  of  the  tangent  of  the 
conic  at  I*  as  being  the  polar  of  I^.  In  this  way  the  formulae 
of  the  last  section  represent  the  polar  of  (-^i  ,71)  even  if  this 
point  lies  on  the  conic. 

There  is  one  other  case  in  which  our  definition  of  a  polar 
breaks  down,  namely  when  P  lies  at  the  centre  of  the  conic  ;  for 
in  this  case  all  chords  through  P  are  bisected  there,  so  that  the 
point  Q  can  never  be  constructed.  T/i£  centre  of  a  cofiic  has  no 
polar. 

lOI.  If  P  lies  on  the  convex  side  of  the  conic,  we  can  draw 
two   tangents   from  it  touching  the  conic  at  P^  and  P^  (v.  the 


172  ANALYTIC    GEOMETRY. 

figure).  In  this  case  the  lineP^P^  is  the  polar  of  P.  For  if 
we  allow  the  secant  P  C  D  \o  turn  around  P  and  approach  the 
position  of  the  tangent  P  P^diS  a.  limit,  both  the  points  C  and  D 
approach  the  point  P^  as  their  limits,  and  since  Q  always  lies 
between  C  and  P>,  it  must  also  approach  P^  as  its  limit. 
Therefore  the  locus  of  Q  points  directly  towards  P^  and,  if 
continued,  passes  through  P^.  In  the  same  way  we  see  that 
the  polar  passes  through  P^  ;  so  that,  being  a  straight  line,  it 
must  coincide  with  the  line  P^  P^. 

102.  We  have  seen  that  every  point  (with  the  single  excep- 
tion of  the  centre  of  the  conic)  has  a  definite  polar.  If  the 
line  A  B  is  the  polar  of  Py  then  conversely  P  is  called  the 
po/e  of  A  B\  and  now  it  is  a  fact  that  every  line  {ivith  the  excep- 
tion of  Hues  through  the  centre  of  the  conic)  has  a  definite  pole. 

In  order  to  prove  this  theorem  let  us  suppose  first  that  our 
conic  is  the  ellipse 

7'  +  !=' (') 

Let  the  given  line  be 

Ax  ^  By  -^  C=  o (2) 

In  order  that  (^lyyi)  should  be  the  pole  of  (2)  it  is  necessary 
and  sufficient  that  its  polar 

^^  +  ^  -  ^  =° (3) 

should  coincide  with  (2),  and  the  condition  for  this  (v.  §  30) 
's  that  the  coefficients  of  (2)  and  (3)  be  proportional  : 

•^1   _       ^        Ji  _        ^ 
Z~~"C'     l?'~~'C' 

Accordingly 

Aa'  Bb"" 


DIAMETERS.       POLES    AND    POLARS.  173 

is  the  pole  of   (2), — a   perfectly  definite  point  if  C  is  not  zero, 
that  is,  if  (2)  does  not  pass  through  the  centre  of  the  ellipse. 

A  similar  proof  holds  in  the  cases  of  the  hyperbola  and  the 
parabola. 

103.  The  most  important  property  of  poles  and  polars  is 
expressed  in  the  following  : 

Theorem. — //  two  points  P^  and  F^  and  their  polars  A^B^ 
and  A^B^  are  given,  and  if  A^B^  passes  through  /\,  then  A^B^ 
will  pass  through  Fi. 

We  will  prove  this  theorem  for  the  case  of  the  ellipse,  lie 
proofs  for  the  hyperbola  or  the  parabola  being  almost  identica,. 

Let  the  coordinates  of  F^  and  F^  be  (.r,,  y,)  and  (^2,  J'J  re- 
spectively.    Then  the  equations  of  A^  B ^  and  A^  B^  are 

■^1  ^  ji_y\y  (  \ 

a'     +  7^  =  '' (') 

"-#■+-^=1 (a) 

a  o 

Since  by  hypothesis  (a:,,  jj  lies  on  (i)  we  have 

d'  "^  b-^  ~  ' 

but  this  is  precisely  the  condition  that  the  point  (a-,,)-,)  should 
lie  on  (2).     This  proves  the  theorem. 

From  tliis  theorem  it  follows  at  once  that  if  several  points 
lie  on  a  line,  their  polars  intersect  on  the  pole  of  that  line,  and  if 
several  lines  tneet  in  a  point,  their  poles  lie  on  the  polar  of  this 
point, 

EXAMPLES. 

I.  Find  the  pole  of  the  line  x  +  ji'  =  i  with  regard 
{a)  to  the  ellipse  x"^  +  ^f  =  i; 
{b)  to  the  parabola/"  =  4  .t. 


I74  ANALYTIC    GEOMETRY. 

2.  Prove  that  the  tangents  at  the  extremity  of  any  focal 
chord  of  a  conic  intersect  on  the  directrix  (v.  Ex.  4,  §  99). 

3.  If  i^  is  a  focus  of  a  conic,  and  /^  is  a  point  on  the  cor- 
responding directrix,  prove  that  the  line  jP  /^  is  perpendicular 
to  the  polar  of  P. 

4.  Show  that  the  fact  established  in  §101  is  a  special  case 
of  the  theorem  of  §  103. 


CHAPTER  IX. 
THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 

104.  We  saw  in  Chapter  III  that  a  straight  line  is  a  curve 
of  the  first  degree,  and,  conversely,  that  every  curve  of  the 
first  degree  is  a  straight  line. 

In  Chapter  VII  we  have  seen  that  the  three  conic  sections 
are  curves  of  the  second  degree.  These,  however,  are  not  the 
only  curves  of  this  kind,  since  two  straight  lines  may  be  rep- 
resented by  a  single  equation  of  the  second  degree  (v.  §  59). 
It  is  our  purpose  in  this  chapter  to  show  that  every  equation 
of  the  second  degree,  if  it  has  a  real  locus,  represents  either 
one  of  the  conic  sections  or  two  straight  lines. 

The  general  equation  of  the  second  degree  is 

Ax""  ^  B xy  ^-  Cy'  +  I?x  +  £y  +  J^  =  o.   .  .  .  (i) 

We  begin  by  examining  the  special  case  of  this  equation   in 
which  B  =z  o: 

Ay+  Cy'  +  £>x  +  £y  +  J^=o (2) 

105.  Before  considering  equation  (2)  of  the  last  section  in 
all  its  generality,  let  us  look  at  two  special  cases. 

Suppose  first  that  D  =  £  =^  o  while  neither  A  nor  C  is  zero. 
The  equation  becomes 

Ax'  +  'Cy'  +  J^=o, (i) 

175 


176  ANALYTIC    GEOxMETRY. 

or,  as  we  may  write  it, 


F 

or,  finally, 


-  ^-^^  -  ^/=  ^' (2) 


t 


A  C 


(3) 


If  tliese  denominators  are  positive  we  may  denote  their  square 
roois  by  a  and  b^  and  (3)  takes  the  standard  form  for  the  equa- 
ticm  of  an  ellipse  (v.  §63,  (7)).  If  the  first  denominator  is 
positive  and  the  second  negative  we  have  in  the  same  way  the 
standard  form  for  the  equation  of  the  hyperbola.  If  the  first 
denominator  is  negative  and  the  second  positive  equation  (3) 
takes  the  form 


^  _/   _  _ 
a         o 


which,  as  we  know  (v.  §  86),  also  represents  a  hyperbola,  but 
one  whose  transverse  axis  lies  on  the  axis  of  Y,  Finally,  if 
the  two  denominators  in  (3)  are  negative,  the  equation  (3)  has 
no  locus  since  its  right-hand  side,  being  essentially  negative, 
cannot  be  equal  to  i.  In  this  case  (3)  is  often  spoken  of  as 
an  imaginary  ellipse. 

In  this  discussion  we  have  tacitly  assumed  that  F  is  not 
zero,  since  in  passing  from  (i)  to  (2)  we  divided  by  F.  If 
F  -=  o  equation  (i)  becomes 

Ax'  +  Cy'  =  o (4) 

[f  A  and  C  have  the  same  sign,  this  equation  is  obviously 
satisfied  only  by  the  values  x  ^=  o,  y  =  o.  The  locus  of  (4)  is 
therefore  a  single  point,  the  origin;  and  (4)  is  spoken  of  as  a 
nu/i  ellipse  (cf.  the   use  of  the  term   null  circle  in  §  46).     \i  A 


THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE.   I  77 

and  C  have  different  signs  Ave  may  suppose  ^  })ositive  and  C 
negative  (as  otherwise  we  could  change  the  signs  throughout  in 
(4)),  and  then  (4)  may  be  written 

a'  x'  -  b'f  =  o, 

an  equation  which  represents  the  two  straight  lines 

ax  —  by  =  o, 

a  X  +  by  =  o, 

which  intersect  at  the  origin. 

Thus  we  have  seen  that,  it  being  assumed  that  neither  ^ 
nor  C  is  zero,  equation  (i)  represents  an  ellipse  (real,  null,  or 
imaginary),  an  hyperbola,  or  two  intersecting  straiglit  lines. 

106.  As  a  second  special  case  of  equation  (2),  §  104,  let  us 
suppose  that  A  =  E  =  o,  while  Cis  not  zero: 

Cy''+Dx  +  F=o (i) 

If  Z>  =  o  this  equation  may  be  written 


^=±/-f (^) 

This  equation  has  no  real  locus  if  C  and  F  have  the  same 
sign.  It  represents  two  lines  parallel  to  the  axis  of  x  if  C  and 
i^  have  opposite  signs;  and  finally  if  F=  o  we  have  no  longer 
two  distinct  lines,  but  only  the  single  linej'  =  o.  These  three 
^ases  may  be  conveniently  characterized  by  saying  we  liave 
two  parallel  lines,  real  and  distinct,  real  and  coincident,  or 
imaginary. 

If,  however,  D  is  not  zero,  we  write  (i)  in  the  form 

cf  =  -r>[x  +  ^, (3) 

and  now  we  make  use  of  a  device,  which  we  shall  have  to  em- 


178  ANALYTIC    GEOMETRY. 

ploy  again  presently.  We  make  the  transformation  of  coordi- 
nates 

that  is  (v.  §  41  (3))  we  shift  our  co5rdinate  axes,  without  turn- 
ing them,  so  that  the  origin  of  the  new  system  lies  at  the  point 

(—  — ,  o  1.  After  this  transformation  of  coordinates  the 
V     ^        / 

equation  of  the  curve  (3)  has  the  form 

/"  =  -f^' (4) 

and  this  is  in  the  standard  form,y  =  2  mx,  for  the  equation 
of  a  parabola.     Accordingly 

Equaiio7i  (i)  represents  either  a  parabola  or  two  parallel 
straight  lines  {distinct,  coincident,  or  imaginary). 

Precisely  the  same  reasoning  applies  to  the  equation 

Ax'  +  Ey  +  F  =  o (5) 

(where  we  assume  that  A  is  not  zero)  except  that  the  x  siwdy 
coordinates  are  interchanged. 

107.  We  are  now  in  a  position  to  take  up  the  consideration 
of  equation  (2),  §  T04.  We  will  first  illustrilte  the  method  to 
be  used  by  a  numerical  example. 

Suppose  we  have  the  equation 

2^'  -  3/ +  4^  +  5J^^  +  3  =  o- 

Let  us  rewrite  this  equation,  taking  first  the  two  terms  in  x, 
then  the  two  terms  in  y,  and  last  the  constant  term.  We  will 
also  take  out  the  factor  2  (the  coefficient  of  x"^)  before  the  two 


THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE.   Ijg 

terms  in  x,  and  the  factor  —  3  (the  coefficient  of  y)  before  the 
two  terms  iny: 

2(x^  +  2x)  -  s  (/  -  I7)  +  3  =  o- 

Let  us  now  complete  the  squares  for  the  two  expressions  in 
parentheses  by  adding  2  to  the  first  term  and  —  f|  to  the 
second. 

2  (^  +  ir  -  3  (^  - 1)^  +  (3- 2  +  fi)  =  o. 

If  we  now  make  the  transformation  of  coordinates 

x'  =  X  -{-   I, 

i.e.,  if  we  shift  the  coordinate  axes  without  turning  so  that  the 
origin  lies  at  the  point  (—  i,  f)  the  equation  of  the  curve  takes 
the  form 

2  x'-'-  3/^  +  ^  =  0. 

This  equation  is  of  the  form  considered  in  §  105.  Following 
the  method  there  indicated  we  throw  it  into  the  form 

^  _  /_'  _  _ 

Accordingly  our  original  equation  represents  an  hyperbola 
whose  centre  is  at  the  point  (—  i,  f),  whose  transverse  axis  is 
parallel  to  the  axis  of  V  and  is  of  length  2^11-  =  i'^^37, 
and  whose  conjugate  axis  is  of  length   2V'||-  =  -^1^222. 

Precisely  the  method  just  explained  applies  to  any  equation 
of  the  form  (2),  §  104,  provided  neither  A  nor  C  is  zero.  We 
first  write  the  equation  in  the  form 


('■ 


|,Uc(/  +  |>.)+i^  =  o. 


l8o  ANALYTIC    GEOMETRY. 

and  then,  by  completing  the  square  in  each  of  the  parentheses, 
in  the  form 


(-^.r-(^-i) 


4A      4C 

The  transformation  of  coordinates 


2  A 


reduces  this  equation  to 

and  this  equation  is  of  the  form  considered  in  §  105. 

108.  If  either  A  or  C  is  zero  the  method  must  be  slightly 
modified.  Suppose,  for  instance,  A  —  o.  Then  we  can  write 
equation  (2),  §  104,  in  the  form 

C^y  +  I^)    ^Dx^  F^o (i) 

Completing  the  square,  this  equation  becomes 

After  the  transformation  of  coordinates 


^  =^  +  7c' 


E    \    (3) 


the  equation  of  the  curve  reduces  to 

Cy"'  +  Dx'  +  {f  -  ^  =0, (4) 

and  tliis  equation  is  of  the  for-.ii  ('),  §  ^06. 


THE   GENERAL  EQUATION  OF  THE  SECOND  DEGREE.   l8l 

Similarly  if  C  —  o  the  equation  can  be  reduced  to  the  form 
(5),  §  io6. 

109.  Thus  we  have  proved  that  if  neither  A  nor  C  is  zero 
equation  (2),  §  104,  represents  an  ellipse  (real,  null,  or  imagin- 
ary), an  hyperbola,  or  two  intersecting  straight  lines;  while  if  A 
or  C  is  zero  it  represents  either  a  parabola  or  two  parallel 
straight  lines  (distinct,  coincident,  or  imaginary). 

Moreover,  the  method  we  have  used  enables  us  in  every  case 
to  determine  not  only  what  kind  of  curve  is  represented  by  the 
equation,  but  just  how  it  is  situated. 

EXERCISES. 

(i.)  What  curves  are  represented  by  the  following  equations  ? 
Draw  diagrams  to  show  how  they  are  situated. 

{a.)   2  jc^  +  3^  —  5  =  o- 
ib.)   2  x""  —  3j;^  +  5  =  o. 
((T.)  /  —  2  .V  —  3  =  o. 
id.)  y""  +  7,x  +  2  —  o, 
ie.)  x"  -  3  J'  +  5  33  0. 

(/.)     2  .t'    +    5 J'"'    —  3  X   +    2JF  —  4  =  O. 

ig.)  9  .V-  +  4j'^  —  18  ^  +  2^y  +  27  =  0. 
ih.)   2  jc^  +  3  .T  —  5  _)'  +  2  =  o. 

(2.)  Show  that  the  condition  that  (2),  §  104,  represents  two 
straight  lines  is 

^ACF  -  CD''  -  A  E'  =  o. 

110.  We  come  now  to  the  general  equation  of  the  second 
degree,  (i),  ?'  104,  where  B  is  not  zero.  Let  us  try  to  simplify 
this    equation    by   turning  the   coordinate    axes    through   the 


l82  ANALYTIC    GEOMETRY. 

angle  ^.     The  formulae  for  this  transformation  (v.  (7),  §  42) 

are 

X  =  x'  cos  ^  —  y'  sin  t9,  )  ,  ^ 

J  =  :r'  sin  ^  +  >■'  cos  -9    ) ^^^ 

Equation  (i),  §  104,  now  takes  the  form 

A'  x"  -^  B'  x'y'  +  a y'-'^D'  x'  +  E' y'  +  F=o,      (2) 
where 

^'  =  y^  cos^  i9  +  ^  sin  ^  cos  ^  +  C  sin'  d, (3) 

B'  =z  -  2  A  sin  ^  cos  ^  4-  ^  cos'  ^ 

—  B  sin"-*  t9  4-  2  C  sin  ^  cos  ^,  .  .  .  (4) 

a  =  A  sin'  ^-  B  sin  ^  cos  -^  4-  C  cos'  ^, (5) 

Z)'  =  i:>  cos  -^  4-  -^  sin  i&, (6) 

Z'  =  -  Z>  sin  0  +  ^  cos  ^ (7) 

The  second  of  these  coefficients  may  be  written  more  simply 
as  follows: 

B'  ^B  cos  2  i9-  (^  -  C)  sin  2  ^ (8) 

Accordingly  B'  will  be  zero  if 

B  co^2^={A  —  C)  sin  2  ^, 
i.e.,  if 

cot  2  t9  =:  — ^— (9) 

If  we  remember  that  as  an  angle  increases  from  o  to  180°  its 
cotangent  decreases  from  4-  00  to  —  00 ,  we  see  that  there  is 
just  one  value  of  2  i9  less  than  180°  which  satisfies  (9),  and 
therefore  just  one  acute  value  of  t9. 

If,  then,  we  turn   the    coordinate    axes    through  the  acute 
angle  ^  determined  1)y  (9),  equation  (1),  §  104,  takes  the  form 

A'  x"  -Y  a y' '  +  D'  x'  4-  E' y'  -V  F  ^  o. 


THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE.   1 83 

This  equation  is  of  the  form  (2),  §  104.  Accordingly  we  have 
proved  that  every  equation  of  the  second  degree  which  has  a 
real  locus  represents  either  a  conic  section  or  two  straight 
lines. 

EXAMPLES, 

(i.)  Determine  what  curves  are  represented  by  the  follow- 
ing equations  and  how  they  are  situated.  Illustrate  your 
answers  by  diagrams: 

(a)  x'  +  4xy  +/  +-5  =  0- 
(^)  x""  +  2  xy  —  /  -\-  1  =  0. 
{c)   2  x""  -\-  4xy  +  2y'^  —  ^x  +  y  =  o. 
(d)  2  x'  —  3  xy  +  2/  -  3  .T  +  2^  —  5  =  o. 
This    last    equation,    like    those   which    precede   it,  can  be 
treated  by  the  method  of  §  no.      Let  us,  however,  shift  the 
coordinate  axes  without  turning  them  so  that  the  origin  lies  at 
the  point  {a,  b). 

X  =  x'  +  a, 

y  ~  y  4-  b. 

Equation  id)  then  takes  the  form 

2  ^'*  —  3  x' y'  +  2  y  +  (4  ^  —  3  /^  —  3)  .t'  —  (3^  —  4^  —  2)y 
+  {2  a"^  —  ^a  b  +  2  F  —  3  ^  +  2  ^  —  5)  =  o. 

Now  a  and  b  can  be  so  chosen  that  the  coefficient  of  x^  and 
y  are  zero: 

a  =  i,     b  =  \. 

Thus  if  we  shift  to  the  new  origin  (f ,  4^),  equation  {d)  takes 
the  form 

14  x^  —  2\  X  y  4-  14  jf''  —  43  =  o, 

and  this  equation  can  be  much  more  easily  treated  than  equa- 
tion {d)  by  the  method  of  §  no. 

(2.)  Discuss  by  the  method  just  explained  the  equation 

3  x:^  -h  5  X y  +  /  +  2  .T  —  3_>'  +  I  =:  o. 


MISCELLANEOUS  EXAMPLES. 

(i.)  Prove  that  the  foot  of  the  perpendicular  from  a  focus 
of  an  hyperbola  to  an  asymptote  is  at  distances  a  and  b  from 
the  centre  and  focus  respectively. 

(2.)  A  circle  has  its  centre  at  the  vertex  (9  of  a  parabola  whose 
focus  is  F^  and  the  diameter  of  the  circle  is  3  (9  F ;  show  that 
the  common  chord  of  the  curves  bisects  OF. 

(3.)  Prove  that  the  semi-minor  axis  of  an  ellipse  is  a  mean 
proportional  between  the  parts  of  a  tangent  at  an  extremity  of 
the  major  axis,  cut  off  by  conjugate  diameters  produced. 

(4.)  Prove  that  the  parts  of  a  normal  to  an  hyperbola  cut  off 
by  the  transverse  and  conjugate  axes  respectively,  are  in  the 
ratio  U^ :  a^. 

(5.)  Prove  that  if  the  major  axis  of  an  ellipse  is  equal  to 
twice  the  minor  axis,  a  straight  line,  equal  to  half  the  major  axis, 
and  which  moves  with  one  end  on  the  upper  half  of  the  curve 
and  the  other  on  the  lower  half  of  the  minor  axis,  is  bisected 
by  the  major  axis. 

(6. )  A  line  of  constant  length  moves  with  its  extremities  on 
two  straight  lines  at  right  angles  to  each  other  :  show  that  the 
locus  of  a  fixed  point  on  the  moving  Une  is  an  ellipse  with  the 
segments  of  the  line  for  semi-axes. 

(7.)  Adapt  the  last  example  to  the  case  in  which  the  fixed 
point  is  taken  on  the  extension  of  the  moving  line. 

(8.)  Two  equal  rulers,  A  B,  B  C,  are  connected  by  a  pivot 
at  B ;  the  extremity  A  is  fixed,  while  C  moves  on  a  given 
straight  line  :  find  the  locus  of  any  fixed  point  of  B  C,  taking 
the  origin  at  A  and  A  C  for  the  axis  of  x. 


MISCELLANLOUS    EXAMPLES.  185 

(9.)  Normals  are  drawn  to  an  ellipse  and  the  circumsciibed 
circle  at  corresponding  points  ;  find  the  locus  of  their  inter- 
section. 

Aus.  A  circle  whose  radius  is  ^  +  ^. 

(10.)  The  ordinate  of  a  point  on  an  hyperbola  is  extended 
until  its  length  equals  a  focal  radius  of  the  point;  find  the 
locus  of  its  extremity. 

(11.)  Any  straight  line  through  the  origin  meets  a  parallel  to 
OX  drawn  through  a  fixed  point,  B^  on  the  axis  of  7,  in  P' ;  on 
OF',F  is  taken  so  that  its  ordinate  equals  B  F' j  find  the 
(ocus  of  F. 

A, 'IS.  A  parabola. 

(12.)  Find  the  locus  of  the  centre  of  a  circle  which  passes 
through  a  fixed  pomt   and   touches   a  given  Ime.     (Axes  as  in 

§  55'  Ex.  13.) 

A71S.    A  parabola. 

(13.)    Given  the  base  and  the  product  of  the  tangents  of  the 

base  angles  of  a  triangle ;  find  the  locus  of  the  vertex. 

Suggestion.     Taking  the  axes  as  in  §  55,  Ex.  5,  the  tangents 

y        J    y 

of  the  base  angles  are  -^ —  and  . 

C  +  X  C  —  X 

A?is.  Letting  m  represent  the  given  product,  the  locus  is  an 
ellipse  or  hyperbola,  according  as  m   is  positive  or  negative. 

(14.)  Given  the  base  and  the  product  of  the  tangents  of  the 
halves  of  the  base  angles  ;  find  the  locus  of  the  vertex. 

If  we  express  the  tangents  of  the  half  angles  in  terms  of  the 
sides,  we  shall  find  that  the  sum  of  the  sides,  2  i"  is  given;  the 
locus  must  therefore  be  an  ellipse  with  the  extremities  of  the 
base  for  its  foci. 

(15.)  Show  by  means  of  the  results  of  the  last  two  examples 
that  when  the  base  and  the  sum  of  the  sides  of  a  triangle  are 
given,  the  centre  of  the  inscribed  circle  moves  on  an  ellipse 
whose  major  axis  is  the  base  of  the  triangle. 


M 


ANALYTIC    GEOMETRY 


(i6.)  Show  that  the  equation  of  the  elHpse  referred  to  the 
major  axis  as  the  axis  of  x,  and  its  left-hand  extremity  as  origin, 
may  be  written  in  the  form 

y^  —  ^^  {2  a  X  —  x^'). 
a" 

(17.)  By  means  of  this  equation,  prove  that  the  locus  of  the 
middle  points  of  chords  of  an  ellipse,  which  pass  through  the 
left-hand  extremity  of  the  major  axis,  is  another  ellipse.  What 
are  its  axes,  and  how  is  it  situated  ? 

(18.)  Prove  that  if  the  right-hand  focus  of  an  ellipse  is  the 
origin  (the  major  axis  being  the  axis  of  x)^  the  radius  connecting 
that  focus  with  any  point  of  the  ellipse  equals  a  {\  —  e^)  —ex; 
and  when  the  left-hand  focus  is  the  origin,  the  radius  joining 
that  focus  with  a  point  of  the  ellipse  \^  a  {\  —  e^)  ^  e  x. 

(19.)  By  means  of  the  results  of  the  last  example,  obtain  the 
polar  equations  of  the  ellipse  referred  to  the  right-hand  and  left- 
hand  focus,  respectively. 


Ans.  r  — 


^  (i  -^')      ^  _  ^  (i  -  g^) 


r  z= 


I  +  e  cos  cfi  I  —  e  cos  <^ 

(20.)  Show  that  for  the  right-hand  branch  of  the  hyperbola 
the  focal  radii,  when  referred  to  the  respective  foci  (v.  Ex.  18), 
are 

e  X  —  a  (i  —  e^)  and  e  x  -\-  a  {\  —  e^)  ; 

changing  the  signs  of  these  expressions,  we  have  the  focal  radii 
for  the  left-hand  branch. 

(21.)  From  the  results  of  the  last  example,  show  that  the 
polar  equations  of  the  right-hand  branch  of  an  hyperbola  re- 
ferred to  the  foci  are 

a  i\  —  e^\  a  {\  —  e'-') 

r  = ^^ ^~   and   r  =  — ^^ i-. 

1  —  e  cos  </)  I  —  e  cos  *f> 

The  negative  values  of  r  derived  from  these  equations  give 


MISCELLANEOUS    EXAMPLES.  I  87 

points  on  the  left-hand  branch,  or  we  may  use  for  that  branch 
the  following  equations,  similarly  obtained  :  — 

a  (i  —  e-)          ,                 a  (i  —  e"^) 
r—  — ^^^ i-   and    r  — ^^ '—. 

I  +  e  cos  (f>  I  +  ^  cos  (/) 

(22.)  Obtain  the  polar  equation  of  the  parabola  referred  to 
the  focus,  by  transformation  of  coordinates. 

A71S.    r  = ^        . 

I  —  cos  cfi 

(23.)  Find  the  polar  equation  mentioned  in  the  last  example 
from  the  definition  of  the  curve,  without  assuming  any  other 
form  of  the  equation  of  the  parabola. 

(24.)  Discuss  the  polar  equation  of  the  parabola  referred  to 
the  focus. 

(25.)  From  the  equation  of  the  parabola  referred  to  axis  and 
directrix,  obtain  the  polar  equation  when  the  intersection  of 
these  lines  is  the  pole. 

(26.)  Find  the  locus  of  the  middle  points  of  lines  drawn  to 
a  parabola  from  (i)  the  vertex,  (2)  the  focus,  (3)  the  intersec- 
tion of  axis  and  directrix.' 

•    Ans.    Parabolas  whose  parameters  are  one  half  that  of  the 
original  curve. 

(27.)  Prove  that  if  a  straight  line  is  drawn  from  the  intersec- 
tion of  the  directrix  and  axis  of  a  parabola  cutting  the  curve,  the 
rectangle  of  the  intercepts  made  by  the  curve  is  equal  to  the 
rectangle  of  the  parts  into  which  the  parallel  focal  chord  is 
divided  by  the  focus. 

(28.)  Two  parabolas  have  the  same  focus  and  the  same 
transverse  axis.  A  tangent  to  the  first  curve  and  a  tangent  to 
the  second  move  so  as  to  remain  at  right  angles  to  each  other. 
Find  the  locus  of  their  intersection. 

Suggestion.     Take  the  common  focus  as  origin. 

A?is.  The  straight  line  parallel  to  the  two  directrices  and 
half  way  between  them. 


ibS  ANALYTIC    GEOMETRY. 

(29.)  Show,  by  the  method  of  §  s;^,  that  the  tangent  to  an 
ellipse  or  hyperbola  at  any  point  makes  equal  angles  with  tlie 
focal  radii  drawn  to  that  point,     (v.  §§  79,  81.) 

(30.)  Prove,  as  in  the  last  example,  that  the  tangent  to  a 
parabola  at  any  point  makes  equal  angles  with  the  axis  and  the 
focal  radius  of  that  point. 

(31.)  Prove  that  tangents  to  the  parabola  at  P^  and  J^^  in- 
tersect at  the  point  whose  coordinates  are 

2  2  m 

(32.)  From  the  results  of  the  last  example,  show  that  the 
area  of  a  triangle  formed  by  three  tangents  to  a  parabola  is  one 
half  the  area  of  the  triangle  whose  vertices  are  the  points  of 
tangency. 

(^S-)  Prove  that  the  triangle  formed  by  a  tangent  to  an  hy- 
perbola and  the  asymptotes  has  a  constant  area. 

(34.)  If  from  any  point  F'  oi  2l  line  M'  F',  perpendicular  to 
the  axis  C  O  M'  of  a  parabola  whose  vertex  is  O,  a  parallel  to 
the  axis  is  drawn  meeting  the  curve  in  F'^ ;  show  that,  if  C  (7  is 
made  equal  to  O  M',  the  locus  of  the  intersection  oi  O  F'  and 
C  F"  is  the  original  curve. 

(35.)  F'  P"  is  any  chord  of  an  ellipse  at  right  angles  to  the 
major  axis  A^  A  ;  find  the  locus  of  the  intersection  of  A^  F'  and 
AFf'. 

Ans.    An  hyperbola  with  the  same  axes  as  the  ellipse. 

(36.)  The  right-hand  extremity  of  a  diameter  of  a  circle  is 
joined  with  the  middle  point  of  a  parallel  chord  ;  find  the  locus 
of  the  intersection  of  this  line  with  the  radius  drawn  to  the  right- 
hand  extremity  of  the  chord. 

Ans.   A  parabola. 

(37.)  Prove  that  the  perpendicular  from  the  centre  of  an  el- 
lipse upon  a  straight  line  which  joins  the  ends  of  perpendicular 


MISCELLANEOUS    EXAMPLES.  189 

diameters    is   of  constant   length.      What   is   the  locus  of   its 

extremity  ? 

(38.)  From  a  point  P*  of  an  ellipse  straight  lines  are  drawn 
to  A^  and  A^  the  extremities  of  the  major  axis,  and  from  A  and 
A^  perpendiculars  are  erected  to  A^  F'  and  A  F' ;  show  that 
the  locus  of  their  intersection  is  another  ellipse,  and  find  its 
axes. 

(39.)  Find  the  locus  of  the  intersection  of  the  ordinate  of 
any  point  of  an  ellipse,  produced,  with  the  perpendicular  from 
the  centre  to  the  tangent  at  that  point. 

(40.)  A  perpendicular  is  drawn  from  a  focus  of  an  ellipse  to 
any  diameter  ;  find  the  locus  of  its  intersection  with  the  con- 
jugate diameter. 

A?is.   A  straight  line  perpendicular  to  the  major  axis. 
(41.)    Tw^o  perpendiculars  are  drawn  from  the  extremities  of  a 
pair  of  conjugate  diameters  of  an  ellipse  to  the  diameter  whose 
equation  is 

y  z=  X  tan  a  ; 

show  that  the  sum  of  the  squares  of  the  perpendiculars  is 
a^  sin'^  a  +  b'^  cos^  a. 

(42.)  Prove  that  the  tangents  to  conjugate  hyperbolas  at  the 
extremities  of  conjugate  diameters  intersect  on  the  asymptotes. 

(43.)  Show  that  the  locus  of  the  intersection  of  tangents  to 
an  ellipse  at  the  extremities  of  conjugate  diameters  is  an  ellipse ; 
find  its  axes. 

(44.)  Show  that  the  equation 

xy  =  c 

represents  a  rectangular  hyperbola  whose  asymptotes  are   the 
coordinate  axes. 


190  ANALYTIC    GEOMETRY. 

(45.)  Obtain  by  the  method  of  §  49  the  equation  of  the 
tangent    to    the    rectangular    hyperbola  xy  =  <r   at   the   point 

(-^i  >  }'i)-  ^^^^'  J\  ^  +  ^xy  =  2  <;. 

(46.)  Obtain  by  the  method  of  §  99  the  equation  of  the 
polar  of  the  point  (^i,J'i)  with  regard  to  the  hyperbola 
X  V  —  c. 

(47.)  Prove  that  if  two  rectangular  hyperbolas  are  so 
situated  that  the  axes  of  one  are  the  asymptotes  of  the  other, 
the  polars  of  any  point  with  regard  to  the  two  hyperbolas  are 
perpendicular  to  each  other.  Hence  show  that  these  hyper- 
bolas intersect  at  right  angles. 

(48.)  Two  concentric  ellipses  have  axes  lying  on  the  same 
lines.  Tangents  are  drawn  to  one  ellipse.  Find  the  locus  of 
their  poles  with  regard  to  the  other. 

(49.)  Prove  that  the  circle  described  on  any  focal  radius 
of  an  ellipse  as  diameter  is  tangent  to  the  circle  described  on 
the  transverse  axis  of  the  ellipse  as  diameter. 

(50.)  State  and  prove  the  proposition  analogous  to  (49)  in 
the  case  of  the  parabola.     Of  the  hyperbola. 

(51.)  A  set  of  lines  terminated  by  two  rectangular  axes 
pass  through  a  fixed  point.  Find  the  equation  of  the  locus 
of  their  middle  points,  and  determine  what  this  equation 
represents.  Ans.  A  rectangular  hyperbola. 

(52.)  In  an  equilateral  hyperbola  show  that  focal  chords 
parallel  to  conjugate  diameters  are  equal. 

(53.)  Parallel  tangents  are  drawn  to  a  set  of  confocal  ellip- 
ses. Find  the  equation  of  the  locus  of  the  points  of  contact, 
and  determine  what  this  equation  represents. 

Ans.  A  rectangular  hyperbola. 

(54.)  Find  by  the  method  of  §  51  the  equation  of  the  diam- 
eter of  the  hyperbola  xy  —  c  which  bisects  the  chords  whose 
slope  is  /^.  Hence  show  that  the  angles  between  any  pair  of 
conjugate  diameters  of  a  rectrngular  hyperbola  are  bisected 
by  the  asymptotes. 


MISCELLANEOUS    EXAMPLES.  I91 

(55.)  Find  by  the  method  of  §  49  the  equation  of  the  tangent 
to  the  conic 

A  x'  +  Bxy  +  Cf  +  nx  +  Ey  +  F  =  0 

at  the  point  Gv,j',). 

/?  Z) 

Ans.  A  x^  X  +  -  {y^  x  +  x^y)  +  Cyj'  +  -{x  +  x) 
2  2 

+  -iy+y.)  +  F=o. 
2 

(56.)  Find  the  polar  of  (.r,  ,j',)  with  regard  to  the  conic  of 
Ex.  (55). 

(57.)  Show  that  the  condition  that  the  conic  of  Ex.  (55)  be 
a  rectangular  hyperbola  is  ^  =  —  C. 


CIYIL  ENGINEERING 

U.  of  C. 

ASSOCIATION  LIBRARY 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


OCT  2  6  . 
v^.'^,3  1  ^  1951 


^     ao'Ji 


28  1952^^ 


AUG 
OCT  20  195 


LD  21-100m-9,'47(A5702sl6)476 


800299 


Lib- 
UNIVERSITY  OF  CALIFORNIA  UBRARY 


